Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study

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1 Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic Functions 2-4 Eponential Functions 2- Logarithmic Functions Chapter 2 Review of mathematics beond the elementar level requires a firm understanding of a basic list of elementar functions, their properties, and their graphs. See the inside front cover of this book for a list of the functions that form our librar of elementar functions. Most functions in the list will be introduced to ou b the end of Chapter 2 and should become a part of our mathematical toolbo for use in this and most future courses or activities that involve mathematics. A few more elementar functions ma be added to these in other courses, but the functions listed inside the front cover are more than sufficient for all the applications in this tet. Review Eercise 46

2 Section 2.1 Functions 47 Section 2-1 FUNCTIONS Equations in Two Variables Definition of a Function Functions Specified b Equations Function Notation Applications We introduce the general notion of a function as a correspondence between two sets. Then we restrict attention to functions for which the two sets are both sets of real numbers. The most useful are those functions that are specified b equations in two variables. We discuss the terminolog and notation associated with functions, graphs of functions, and applications to economics. Equations in Two Variables In Chapter 1 we found that the graph of an equation of the form A + B = C, where A and B are not both zero, is a line. Because a line is determined b an two of its points, such an equation is eas to graph: Just plot an two points in its solution set and sketch the unique line through them. More complicated equations in two variables, for eample, = 9-2 or 2 = 4, are more difficult to graph.to sketch the graph of an equation, we plot enough points from its solution set in a rectangular coordinate sstem so that the total graph is apparent and then connect these points with a smooth curve. This process is called pointb-point plotting. EXAMPLE 1 SOLUTIONS Point-b-Point Plotting Sketch the graph of each equation. (A) = 9-2 (B) 2 = 4 (A) Make up a table of solutions that is, ordered pairs of real numbers that satisf the given equation. For eas mental calculation, choose integer values for After plotting these solutions, if there are an portions of the graph that are unclear, plot additional points until the shape of the graph is apparent. Then join all the plotted points with a smooth curve as shown in Figure 1.Arrowheads are ( 1, 8) ( 2, ) (0, 9) (1, 8) (2, ) ( 3, 0) (3, 0) ( 4, 7) (4, 7) 9 2 FIGURE 1 = 9-2

3 48 CHAPTER 2 Functions and Graphs 2 4 used to indicate that the graph continues beond the portion shown here with no significant changes in shape. (B) Again we make a table of solutions here it ma be easier to choose integer values for and calculate values for. Note, for eample, that if = 2, then = ;4; that is, the ordered pairs (4, 2) and (-4, 2) are both in the solution set. FIGURE 2 2 = 4 ;9 ;4 ;1 0 ;1 ;4 ; We plot these points and join them with a smooth curve (Fig. 2). MATCHED PROBLEM 1 Sketch the graph of each equation. (A) = 2-4 (B) 2 = Eplore & Discuss 1 To graph the equation = , we use point-b-point plotting to obtain (A) (B) FIGURE 3 (A) Do ou think this is the correct graph of the equation? If so, wh? If not, wh? (B) Add points on the graph for = -2, -1., -0., 0., 1., and 2. (C) Now, what do ou think the graph looks like? Sketch our version of the graph, adding more points as necessar. (D) Graph this equation on a graphing calculator and compare it with our graph from part (C). The icon in the margin is used throughout this book to identif optional graphing calculator activities that are intended to give ou additional insight into the concepts under discussion. You ma have to consult the manual for our graphing calculator for the details necessar to carr out these activities. For eample, to graph the equation in Eplore Discuss 1 on most graphing calculators, ou first have to enter the equation (Fig. 3A) and the window variables (Fig. 3B). As Eplore Discuss 1 illustrates, the shape of a graph ma not be apparent from our first choice of points on the graph. Using point-b-point plotting, it ma be difficult to find points in the solution set of the equation, and it ma be difficult to determine when ou have found enough points to understand the shape of the graph. We will supplement the technique of point-b-point plotting with a detailed analsis

4 Section 2.1 Functions 49 of several basic equations, giving ou the abilit to sketch graphs with accurac and confidence. Definition of a Function Central to the concept of function is correspondence. You have alread had eperiences with correspondences in dail living. For eample, To each person there corresponds an annual income. To each item in a supermarket there corresponds a price. To each student there corresponds a grade-point average. To each da there corresponds a maimum temperature. For the manufacture of items there corresponds a cost. For the sale of items there corresponds a revenue. To each square there corresponds an area. To each number there corresponds its cube. One of the most important aspects of an science is the establishment of correspondences among various tpes of phenomena. Once a correspondence is known, predictions can be made. A cost analst would like to predict costs for various levels of output in a manufacturing process; a medical researcher would like to know the correspondence between heart disease and obesit; a pschologist would like to predict the level of performance after a subject has repeated a task a given number of times; and so on. What do all the eamples above have in common? Each describes the matching of elements from one set with the elements in a second set. Consider the tables of the cube, square, and square root given in Tables 1 3. Tables 1 and 2 specif functions, but Table 3 does not. Wh not? The definition of the term function will eplain. TABLE 1 TABLE 2 TABLE 3 Domain Range Domain Range Domain Range Number Cube Number Square Number Square root DEFINITION Function A function is a correspondence between two sets of elements such that to each element in the first set there corresponds one and onl one element in the second set. The first set is called the domain, and the set of corresponding elements in the second set is called the range. Tables 1 and 2 specif functions, since to each domain value there corresponds eactl one range value (for eample, the cube of -2 is -8 and no other number). On the other hand,table 3 does not specif a function, since to at least one domain value there corresponds more than one range value (for eample, to the domain value 9 there corresponds -3 and 3, both square roots of 9).

5 0 CHAPTER 2 Functions and Graphs Eplore & Discuss 2 Consider the set of students enrolled in a college and the set of facult members of that college. Suppose we define a correspondence between the two sets b saing that a student corresponds to a facult member if the student is currentl enrolled in a course taught b that facult member. Is this correspondence a function? Discuss. FIGURE 4 1 Functions Specified b Equations Most of the functions in this book will have domains and ranges that are (infinite) sets of real numbers. The graph of such a function is the set of all points (, ) in the Cartesian plane such that is an element of the domain and is the corresponding element in the range. The correspondence between domain and range elements is often specified b an equation in two variables. Consider, for eample, the equation for the area of a rectangle with width 1 inch less than its length (Fig. 4). If is the length, then the area is given b = ( - 1) Ú 1 For each input (length), we obtain an output (area). For eample, If =, then = ( - 1) = # 4 = 20. If = 1, then = 1 (1-1) = 1 # 0 = 0. If = 1, then = 1 (1-1) = - 1 L The input values are domain values, and the output values are range values. The equation assigns each domain value a range value. The variable is called an independent variable (since values can be independentl assigned to from the domain), and is called a dependent variable (since the value of depends on the value assigned to ). In general, an variable used as a placeholder for domain values is called an independent variable; an variable that is used as a placeholder for range values is called a dependent variable. When does an equation specif a function? DEFINITION Functions Specified b Equations If in an equation in two variables, we get eactl one output (value for the dependent variable) for each input (value for the independent variable), then the equation specifies a function. The graph of such a function is just the graph of the specifing equation. If we get more than one output for a given input, the equation does not specif a function. EXAMPLE 2 Functions and Equations Determine which of the following equations specif functions with independent variable. SOLUTION (A) 4-3 = 8, a real number (B) 2-2 = 9, a real number (A) Solving for the dependent variable, we have 4-3 = 8 4 = = (1)

6 Section 2.1 Functions 1 Since each input value corresponds to eactl one output value we see that equation (1) specifies a function. (B) Solving for the dependent variable, we have ( = ), 2-2 = 9 2 = (2) 2 = ;29 + Since is alwas a positive real number for an real number and since each positive real number has two square roots,* to each input value there corresponds two output values ( = and = ). For eample, if = 4, then equation (2) is satisfied for = and for = -. Thus, equation (2) does not specif a function. MATCHED PROBLEM 2 Determine which of the following equations specif functions with independent variable. (A) 2-4 = 9, a real number (B) 3-2 = 3, a real number Since the graph of an equation is the graph of all the ordered pairs that satisf the equation, it is ver eas to determine whether an equation specifies a function b eamining its graph. The graphs of the two equations we considered in Eample 2 are shown in Figure. (A) (B) FIGURE In Figure A notice that an vertical line will intersect the graph of the equation 4-3 = 8 in eactl one point. This shows that to each value there corresponds eactl one value and confirms our conclusion that this equation specifies a function. On the other hand, Figure B shows that there eist vertical lines that intersect the graph of 2-2 = 9 in two points. This indicates that there eist values to which there correspond two different values and verifies our conclusion that this equation does not specif a function. These observations are generalized in Theorem 1. THEOREM 1 VERTICAL-LINE TEST FOR A FUNCTION An equation specifies a function if each vertical line in the coordinate sstem passes through at most one point on the graph of the equation. If an vertical line passes through two or more points on the graph of an equation, then the equation does not specif a function. * Recall that each positive real number N has two square roots: 2N, the principal square root, and - 2N, the negative of the principal square root (see Appendi A, Section A-6).

7 2 CHAPTER 2 Functions and Graphs The function graphed in Figure A is an eample of a linear function.the verticalline test implies that equations of the form = m + b, where m Z 0, specif functions; the are called linear functions. Similarl, equations of the form = b specif functions; the are called constant functions, and their graphs are horizontal lines. The vertical-line test implies that equations of the form = a do not specif functions; note that the graph of = a is itself a vertical line. Eplore & Discuss 3 The definition of a function specifies that to each element in the domain there corresponds one and onl one element in the range. (A) Give an eample of a function such that to each element of the range there correspond eactl two elements of the domain. (B) Give an eample of a function such that to each element of the range there corresponds eactl one element of the domain. In Eample 2, the domains were eplicitl stated along with the given equations. In man cases, this will not be done. Unless stated to the contrar, we shall adhere to the following convention regarding domains and ranges for functions specified b equations: If a function is specified b an equation and the domain is not indicated, then we assume that the domain is the set of all real number replacements of the independent variable (inputs) that produce real values for the dependent variable (outputs). The range is the set of all outputs corresponding to input values. In man applied problems the domain is determined b practical considerations within the problem (see Eample 7). EXAMPLE 3 SOLUTION Finding a Domain Find the domain of the function specified b the equation = 14 -, assuming that is the independent variable. For to be real, 4 - must be greater than or equal to 0; that is, 4 - Ú 0 - Ú -4 4 Sense of inequalit reverses when both sides are divided b 1. Thus, Domain: 4 (inequalit notation) or (-q, 4] (interval notation) MATCHED PROBLEM 3 Find the domain of the function specified b the equation = 2-2, assuming is the independent variable. Function Notation We have just seen that a function involves two sets, a domain and a range, and a correspondence that assigns to each element in the domain eactl one element in the range. We use different letters to denote names for numbers; in essentiall the same wa, we will now use different letters to denote names for functions. For eample, f and g ma be used to name the functions specified b the equations = and = : f: = g: = (3)

8 Section 2.1 Functions 3 f If represents an element in the domain of a function f, then we frequentl use the smbol DOMAIN FIGURE 6 f () RANGE f () in place of to designate the number in the range of the function f to which is paired (Fig. 6). This smbol does not represent the product of f and. The smbol f() is read as f of, f at, or the value of f at. Whenever we write = f(), we assume that the variable is an independent variable and that both and f() are dependent variables. Using function notation, we can now write functions f and g in (3) in the form f() = and g() = Let us find f(3) and g(-). To find f(3), we replace with 3 wherever occurs in f() = and evaluate the right side: f() = f(3) = 2 # = = 7 For input 3, the output is 7. Thus, f(3) = 7 The function f assigns the range value 7 to the domain value 3. To find g(-), we replace each b - in g() = and evaluate the right side: Thus, g() = g(-) = (-) 2 + 2(-) - 3 = = 12 For input, the output is 12. g(-) = 12 The function g assigns the range value 12 to the domain value. It is ver important to understand and remember the definition of f(): For an element in the domain of the function f, the smbol f() represents the element in the range of f corresponding to in the domain of f. If is an input value, then f() is the corresponding output value. If is an element that is not in the domain of f, then f is not defined at and f() does not eist. EXAMPLE 4 Function Evaluation If 12 f() = g() = 1-2 h() = then * (A) = 12 f(6) = = 3 (B) g(-2) = 1 - (-2) 2 = 1-4 = -3 (C) h(-2) = = 1-3 But 1-3 is not a real number. Since we have agreed to restrict the domain of a function to values of that produce real values for the function, -2 is not in the domain of h and h(-2) does not eist. (D) f(0) + g(1) - h() = (1-12 ) = = -6-3 = -9 * Dashed boes are used throughout the book to represent steps that are usuall performed mentall.

9 4 CHAPTER 2 Functions and Graphs MATCHED PROBLEM 4 Use the functions in Eample 3 to find (A) f(-2) (B) g(-1) (C) h(-8) (D) f(3) h() EXAMPLE SOLUTION Finding Domains Find the domains of functions f, g, and h: f() = 12-2 g() = 1-2 h() = 2-1 Domain of f: 12>( - 2) represents a real number for all replacements of b real numbers ecept for = 2 (division b 0 is not defined). Thus, f(2) does not eist, and the domain of f is the set of all real numbers ecept 2.We often indicate this b writing f() = 12-2 Z 2 Domain of g: The domain is R, the set of all real numbers, since 1-2 represents a real number for all replacements of b real numbers. Domain of h: The domain is the set of all real numbers such that 2-1 is a real number that is, such that - 1 Ú 0 Ú 1 or 31, q ) MATCHED PROBLEM Find the domains of functions F, G, and H: F() = G() = + 3 H() = 22 - In addition to evaluating functions at specific numbers, it is important to be able to evaluate functions at epressions that involve one or more variables. For eample, the difference quotient f( + h) - f() h is studied etensivel in calculus. INSIGHT and h in the domain of f, h Z 0 In algebra, ou learned to use parentheses for grouping variables. For eample, 2( + h) = 2 + 2h Now we are using parentheses in the function smbol f(). For eample, if f() = 2, then f( + h) = ( + h) 2 = 2 + 2h + h 2 Note that f() + f(h) = 2 + h 2 Z f( + h). That is, the function name f does not distribute across the grouped variables ( + h) as the 2 does in 2( + h) (see Appendi A, Section A-2). Eplore & Discuss 4 Let and h be real numbers. (A) If f() = 4 + 3, which of the following is true? 1. f( + h) = h 2. f( + h) = 4 + 4h f( + h) = 4 + 4h + 6

10 (B) If g() = 2, which of the following is true? g( + h) = 2 + h g( + h) = 2 + h 2 g( + h) = 2 + 2h + h 2 Section 2.1 Functions (C) If M() = , describe the operations that must be performed to evaluate M( + h). EXAMPLE 6 SOLUTION Using Function Notation For f() = , find f(a + h) - f(a) (A) f(a) (B) f(a + h) (C) f(a + h) - f(a) (D), h Z 0 h (A) (B) (C) (D) f(a) = a 2-2a + 7 f(a + h) = (a + h) 2-2(a + h) + 7 = a 2 + 2ah + h 2-2a - 2h + 7 f(a + h) - f(a) = (a 2 + 2ah + h 2-2a - 2h + 7) - (a 2-2a + 7) f(a + h) - f(a) h = 2ah + h 2-2h = 2ah + h2-2h h = 2a + h - 2 = h(2a + h - 2) Because h Z 0, h h h = 1. MATCHED PROBLEM 6 Repeat Eample 6 for f() = APPLICATIONS We now turn to the important concepts of break-even and profit loss analsis, which we will return to a number of times in this book. An manufacturing compan has costs, C, and revenues, R. The compan will have a loss if R 6 C, will break even if R = C, and will have a profit if R 7 C. Costs include fied costs such as plant overhead, product design, setup, and promotion; and variable costs, which are dependent on the number of items produced at a certain cost per item. In addition, price demand functions, usuall established b financial departments using historical data or sampling techniques, pla an important part in profit loss analsis. We will let, the number of units manufactured and sold, represent the independent variable. Cost functions, revenue functions, profit functions, and price demand functions are often stated in the following forms, where a, b, m, and n are constants determined from the contet of a particular problem: Cost Function Price Demand Function p = m - n is the number of items that can be sold at $p per item. Revenue Function Profit Function C = (fied costs) + (variable costs) = a + b R = (number of items sold) * (price per item) = p = (m - n) P = R - C = (m - n) - (a + b)

11 6 CHAPTER 2 Functions and Graphs Eample 7 and Matched Problem 7 eplore the relationships among the algebraic definition of a function, the numerical values of the function, and the graphical representation of the function. The interpla among algebraic, numeric, and graphic viewpoints is an important aspect of our treatment of functions and their use. In Eample 7, we also see how a function can be used to describe data from the real world, a process that is often referred to as mathematical modeling. The material in this eample will be returned to in subsequent sections so that we can analze it in greater detail and from different points of view. EXAMPLE 7 Price Demand and Revenue Modeling A manufacturer of a popular digital camera wholesales the camera to retail outlets throughout the United States. Using statistical methods, the financial department in the compan produced the price demand data in Table 4, where p is the wholesale price per camera at which million cameras are sold. Notice that as the price goes down, the number sold goes up. TABLE 4 Price Demand (Millions) p ($) TABLE Revenue (Millions) R() (Millions $) Using special analtical techniques (regression analsis), an analst arrived at the following price demand function that models the Table 4 data: p() = (A) Plot the data in Table 4.Then sketch a graph of the price demand function in the same coordinate sstem. (B) What is the compan s revenue function for this camera, and what is the domain of this function? (C) Complete Table, computing revenues to the nearest million dollars. (D) Plot the data in Table.Then sketch a graph of the revenue function using these points. (E) Plot the revenue function on a graphing calculator. () SOLUTION (A) 0 p() Price per camera ($) 0 1 Million cameras FIGURE 7 Price demand In Figure 7, notice that the model approimates the actual data in Table 4, and it is assumed that it gives realistic and useful results for all other values of between 1 million and 1 million.

12 Section 2.1 Functions 7 (B) (C) R() = p() = ( ) million dollars Domain: 1 1 [Same domain as the price demand function, equation ().] TABLE Revenue (Millions) R() (Million $) (D) R() (E) Million dollars Million cameras MATCHED PROBLEM 7 The financial department in Eample 6, using statistical techniques, produced the data in Table 6, where C() is the cost in millions of dollars for manufacturing and selling million cameras. TABLE 6 Cost Data (Millions) C() (Million $) Using special analtical techniques (regression analsis), an analst produced the following cost function to model the data: C() = (6) (A) Plot the data in Table 6. Then sketch a graph of equation (6) in the same coordinate sstem. (B) What is the compan s profit function for this camera, and what is its domain? (C) Complete Table 7, computing profits to the nearest million dollars. TABLE 7 Profit (Millions) P() (Million $)

13 8 CHAPTER 2 Functions and Graphs (D) Plot the points from part (C).Then sketch a graph of the profit function through these points. (E) Plot the profit function on a graphing calculator. Answers to Matched Problems 1. (A) (B) (A) Does not specif a function (B) Specifies a function 3. Ú 2 (inequalit notation) or 32, q) (interval notation) 4. (A) -3 (B) 0 (C) Does not eist (D) 6. Domain of F: R; domain of G: all real numbers ecept -3; domain of H: 2 (inequalit notation) or (- q, 2] (interval notation) 6. (A) a 2-4a + 9 (B) a 2 + 2ah + h 2-4a - 4h + 9 (C) 2ah + h 2-4h (D) 2a + h (A) C() 00 Million dollars Million cameras (B) (C) P() = R() - C() = ( ) - ( ); domain: 1 1 TABLE 7 Profit (Millions) P() (Million $)

14 Section 2.1 Functions 9 (D) P() (E) Million dollars Eercise Million cameras A In Problems 1 8, use point-b-point plotting to sketch the graph of each equation. 1. = = = 2 4. = 2. = 3 6. = 3 7. = = 12 Indicate whether each table in Problems 9 14 specifies a function. 9. Domain Range Domain Range Domain Range Domain Range Domain Domain Range Range Indicate whether each graph in Problems 1 20 specifies a function In Problems 21 30, each equation specifies a function. Determine whether the function is linear, constant, or neither. 21. = = p 23. = = (4 - ) = = ( - 6) = = = 2 + (1 - )(1 + ) + 1 = In Problems 31 and 32 which of the indicated correspondences define functions? Eplain. 31. Let P be the set of residents of Pennslvania and let R and S be the set of members of the U.S. House of Representatives and the set of members of the U.S. Senate, respectivel, elected b the residents of Pennslvania.

15 60 CHAPTER 2 Functions and Graphs (A) A resident corresponds to the congressperson representing the resident s congressional district. (B) A resident corresponds to the senator representing the resident s state. 32. Let P be the set of patients in a hospital, let D be the set of doctors on the hospital staff, and N be the set of nurses on the hospital staff. (A) A patient corresponds to the doctor if that doctor admitted the patient to the hospital. (B) A patient corresponds to the nurse if that nurse cares for the patient. In Problems 33 40, use point-b-point plotting to sketch the graph of each function. 33. f() = f() = f() = f() = f() = f() = f() = f() = -6 In Problems 41 and 42, the three points in the table are on the graph of the indicated function f. Do these three points provide sufficient information for ou to sketch the graph of = f()? Add more points to the table until ou are satisfied that our sketch is a good representation of the graph of = f() on the interval [-, ] f() f() f() = f() = Let f() = 0-2 and g() = + 20, (A) Evaluate f(), g(), and f() - g() for = 0,,, 1, 20. (B) Graph = f(), = g(), and = f() - g() on the interval [0, 20]. 44. Repeat Problem 43 for f() = and g() = In Problems 4 2, use the following graph of a function f to determine or to the nearest integer, as indicated. Some problems ma have more than one answer. f() f() B 4. = f(-) = f() = f() = f() 2. If f() = 2-3 and g() = 2 + 2, find each of the epressions in Problems f(2) 4. f(1). f(-1) 6. g(1) 7. g(-3) 8. g(-2) 9. f(1) + g(2) 60. f(3) - g(3) 61. g(3) # f(0) 62. g(0) # f(-2) 63. g(-2) g(-3) 64. f(-2) f(2) In Problems 6 70, find the domain of each function. 6. F() = H() = f() = g() = g() = F() = Two people are discussing the function f() = = f(4) = f(-2) 3 = f(), = f() and one sas to the other, f(2) eists but f(3) does not. Eplain what the are talking about. 72. Referring to the function in Problem 71, do f(-2) and f(-3) eist? Eplain. The verbal statement function f multiplies the square of the domain element b 3 and then subtracts 7 from the result and the algebraic statement f() = define the same function. In Problems 73 76, translate each verbal definition of a function into an algebraic definition. 73. Function g subtracts from twice the cube of the domain element. 74. Function f multiplies the domain element b -3 and adds 4 to the result. 7. Function G multiplies the square root of the domain element b 2 and subtracts the square of the domain element from the result. 76. Function F multiplies the cube of the domain element b -8 and adds 3 times the square root of 3 to the result. In Problems 77 80, translate each algebraic definition of the function into a verbal definition. 77. f() = g() = F() = G() = 41-2 Determine which of the equations in Problems specif functions with independent variable. For those that do, find

16 the domain. For those that do not, find a value of to which there corresponds more than one value of = = = = = = = = = = If F(t) = 4t + 7, find F(3 + h) - F(3) 92. If G(r) = 3 - r, find G(2 + h) - G(2) 93. If Q() = , find Q(2 + h) - Q(2) h h h 94. If P() = , find P(3 + h) - P(3) h If f() = 2-1, find and simplif each epression in Problems f() 96. f(-3) 97. f(2 + ) 98. f(3-6) 99. f(2) + f() 0. f(3) - f(6) 1. f(f(1)) 2. f(f(-2)) 3. f(2) 4. f(-3). f( + 1) 6. f(1 - ) C Section 2.1 Functions 61 In Problems 7 112, find and simplif each of the following. (A) (B) (C) 7. f( + h) f( + h) - f() f( + h) - f() h f() = f() = f() = f() = f() = (20 - ) 112. f() = ( + 40) Problems , refer to the area A and perimeter P of a rectangle with length l and width w (see the figure). A lw P 2l 2w l 113. The area of a rectangle is 2 square inches. Epress the perimeter P(w) as a function of the width w, and state the domain of this function The area of a rectangle is 81 square inches. Epress the perimeter P(l) as a function of the length l, and state the domain of this function. 11. The perimeter of a rectangle is 0 meters. Epress the area A(l) as a function of the length l, and state the domain of this function The perimeter of a rectangle is 160 meters. Epress the area A(w) as a function of the width w, and state the domain of this function. w Applications 117. Price demand. A compan manufactures memor chips for microcomputers. Its marketing research department, using statistical techniques, collected the data shown in Table 8, where p is the wholesale price per chip at which million chips can be sold. Using special analtical techniques (regression analsis), an analst produced the following price demand function to model the data: p() = TABLE 8 Price Demand (Millions) p ($) Plot the data points in Table 8, and sketch a graph of the price demand function in the same coordinate sstem. What would be the estimated price per chip for a demand of 7 million chips? For a demand of 11 million chips? 118. Price demand. A compan manufactures notebook computers. Its marketing research department, using statistical techniques, collected the data shown in Table 9, where p is the wholesale price per computer at which thousand computers can be sold. Using special analtical techniques (regression analsis), an analst produced the following price demand function to model the data: TABLE 9 p() = 2, Price Demand (Thousands) p ($) 1 1, , , Plot the data points in Table 9, and sketch a graph of the price demand function in the same coordinate sstem. What would be the estimated price per computer for a demand of 11 thousand computers? For a demand of 18 thousand computers?

17 62 CHAPTER 2 Functions and Graphs 119. Revenue. (A) Using the price demand function p() = from Problem 117, write the compan s revenue function and indicate its domain. (B) Complete Table, computing revenues to the nearest million dollars. TABLE Revenue (Millions) R() (Million $) (C) Plot the points from part (B) and sketch a graph of the revenue function through these points. Choose millions for the units on the horizontal and vertical aes Revenue. (A) Using the price demand function p() = 2, from Problem 118, write the compan s revenue function and indicate its domain. (B) Complete Table 11, computing revenues to the nearest thousand dollars. (C) Plot the points in part (B) and sketch a graph of the profit function through these points Profit. The financial department for the compan in Problems 118 and 120 established the following cost function for producing and selling thousand notebook computers: C() = 4, thousand dollars (A) Write a profit function for producing and selling thousand notebook computers, and indicate the domain of this function. (B) Complete Table 13, computing profits to the nearest thousand dollars. TABLE 13 Profit (Thousands) P() (Thousand $) 1 2, (C) Plot the points in part (B) and sketch a graph of the profit function through these points Packaging. A cand bo is to be made out of a piece of cardboard that measures 8 b 12 inches. Equal-sized squares inches on a side will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular bo. TABLE 11 Revenue (Thousands) R() (Thousand $) 1 1, (C) Plot the points from part (B) and sketch a graph of the revenue function through these points. Choose thousands for the units on the horizontal and vertical aes Profit. The financial department for the compan in Problems 117 and 119 established the following cost function for producing and selling million memor chips: C() = million dollars (A) Write a profit function for producing and selling million memor chips, and indicate its domain. (B) Complete Table 12, computing profits to the nearest million dollars. TABLE 12 Profit (Millions) P() (Million $) (A) Epress the volume of the bo V() in terms of. (B) What is the domain of the function V (determined b the phsical restrictions)? (C) Complete Table 14. TABLE 14 Volume V() (D) Plot the points in part (C) and sketch a graph of the volume function through these points Packaging. Refer to Problem 123. (A) Table 1 shows the volume of the bo for some values of between 1 and 2. Use these values to estimate to one TABLE 1 Volume V()

18 Section 2.2 Elementar Functions: Graphs and Transformations 63 decimal place the value of between 1 and 2 that would produce a bo with a volume of 6 cubic inches. (B) Describe how ou could refine this table to estimate to two decimal places. (C) Carr out the refinement ou described in part (B) and approimate to two decimal places. 12. Packaging. Refer to Problems 123 and 124. (A) Eamine the graph of V() from Problem 123D and discuss the possible locations of other values of that would produce a bo with a volume of 6 cubic inches. Construct a table like Table 1 to estimate an such value to one decimal place. (B) Refine the table ou constructed in part (A) to provide an approimation to two decimal places Packaging. A parcel deliver service will onl deliver packages with length plus girth (distance around) not eceeding 8 inches. A rectangular shipping bo with square ends inches on a side is to be used. Length Girth (A) If the full 8 inches is to be used, epress the volume of the bo V() in terms of. (B) What is the domain of the function V (determined b the phsical restrictions)? (C) Complete Table 16. TABLE Volume V() (D) Plot the points in part (C) and sketch a graph of the volume function through these points Muscle contraction. In a stud of the speed of muscle contraction in frogs under various loads, noted British biophsicist and Nobel Prize winner A. W. Hill determined that the weight w (in grams) placed on the muscle and the speed of contraction v (in centimeters per second) are approimatel related b an equation of the form (w + a)(v + b) = c where a, b, and c are constants. Suppose that for a certain muscle, a = 1, b = 1, and c = 90. Epress v as a function of w. Find the speed of contraction if a weight of 16 grams is placed on the muscle Politics. The percentage s of seats in the House of Representatives won b Democrats and the percentage v of votes cast for Democrats (when epressed as decimal fractions) are related b the equation v - 2s = s 6 1, v (A) Epress v as a function of s, and find the percentage of votes required for the Democrats to win 1% of the seats. (B) Epress s as a function of v, and find the percentage of seats won if Democrats receive 1% of the votes. Section 2-2 ELEMENTARY FUNCTIONS: GRAPHS AND TRANSFORMATIONS A Beginning Librar of Elementar Functions Vertical and Horizontal Shifts Reflections, Stretches, and Shrinks Piecewise-Defined Functions The functions g() = 2-4 h() = ( - 4) 2 k() = -4 2 all can be epressed in terms of the function f() = 2 as follows: g() = f() - 4 h() = f( - 4) k() = -4f() In this section we will see that the graphs of functions g, h, and k are closel related to the graph of function f. Insight gained b understanding these relationships will help us analze and interpret the graphs of man different functions.

19 64 CHAPTER 2 Functions and Graphs A Beginning Librar of Elementar Functions As ou progress through this book, and most an other mathematics course beond this one, ou will repeatedl encounter a relativel small list of elementar functions. We will identif these functions, stud their basic properties, and include them in a librar of elementar functions (see the inside front cover). This librar will become an important addition to our mathematical toolbo and can be used in an course or activit where mathematics is applied. We begin b placing si basic functions in our librar. DEFINITION Basic Elementar Functions f() = Identit function h() = 2 Square function m() = 3 Cube function n() = 1 Square root function p() = 1 3 Cube root function g() = ƒ ƒ Absolute value function These elementar functions can be evaluated b hand for certain values of and with a calculator for all values of for which the are defined. EXAMPLE 1 Evaluating Basic Elementar Functions Evaluate each basic elementar function at (A) = 64 (B) = Round an approimate values to four decimal places. SOLUTION (A) f(64) = 64 h(64) = 64 2 = 4,096 m(64) = 64 3 = 262,144 n(64) = 264 = 8 p(64) = = 4 g(64) = ƒ 64 ƒ = 64 Use a calculator. Use a calculator. (B) f(-12.7) = h(-12.7) = (-12.7) 2 = m(-12.7) = (-12.7) 3 L -2, n(-12.7) = p(-12.7) = L g(-12.7) = ƒ ƒ = 12.7 Use a calculator. Use a calculator. Not a real number. Use a calculator. MATCHED PROBLEM 1 Evaluate each basic elementar function at (A) = 729 (B) = -.2 Round an approimate values to four decimal places.

20 Section 2.2 Elementar Functions: Graphs and Transformations 6 REMARK Most computers and graphing calculators use ABS() to represent the absolute value function. The following representation can also be useful: ƒ ƒ = 2 2 Figure 1 shows the graph, range, and domain of each of the basic elementar functions. f() h() m() (A) Identit function f() Domain: R Range: R (B) Square function h() 2 Domain: R Range: [0, ) (C) Cube function m() 3 Domain: R Range: R n() p() g() (D) Square root function n() Domain: [0, ) Range: [0, ) (E) Cube root function 3 p() Domain: R Range: R (F) Absolute value function g() Domain: R Range: [0, ) FIGURE 1 Some basic functions and their graphs Note: Letters used to designate these functions ma var from contet to contet; R is the set of all real numbers. INSIGHT Absolute Value In beginning algebra, absolute value is often interpreted as distance from the origin on a real number line (see Appendi A, Section A-1). distance 6 ( 6) distance 0 If 6 0, then - is the positive distance from the origin to and if 7 0, then is the positive distance from the origin to. Thus, - if 6 0 ƒ ƒ = e if Ú 0 Vertical and Horizontal Shifts If a new function is formed b performing an operation on a given function, then the graph of the new function is called a transformation of the graph of the original function. For eample, graphs of both = f() + k and = f( + h) are transformations of the graph of = f().

21 66 CHAPTER 2 Functions and Graphs Eplore & Discuss 1 Let f() = 2. (A) Graph = f() + k for k = -4, 0, and 2 simultaneousl in the same coordinate sstem. Describe the relationship between the graph of = f() and the graph of = f() + k for k an real number. (B) Graph = f( + h) for h = -4, 0, and 2 simultaneousl in the same coordinate sstem. Describe the relationship between the graph of = f() and the graph of = f( + h) for h an real number. EXAMPLE 2 Vertical and Horizontal Shifts (A) How are the graphs of = ƒ ƒ + 4 and = ƒ ƒ - related to the graph of = ƒ ƒ? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. (B) How are the graphs of = ƒ + 4 ƒ and = ƒ - ƒ related to the graph of = ƒ ƒ? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. SOLUTION (A) The graph of = ƒ ƒ + 4 is the same as the graph of = ƒ ƒ shifted upward 4 units, and the graph of = ƒ ƒ - is the same as the graph of = ƒ ƒ shifted downward units. Figure 2 confirms these conclusions. [It appears that the graph of = f() + k is the graph of = f() shifted up if k is positive and down if k is negative.] (B) The graph of = ƒ + 4 ƒ is the same as the graph of = ƒ ƒ shifted to the left 4 units, and the graph of = ƒ - ƒ is the same as the graph of = ƒ ƒ shifted to the right units. Figure 3 confirms these conclusions. [It appears that the graph of = f( + h) is the graph of = f() shifted right if h is negative and left if h is positive the opposite of what ou might epect.] 4 4 FIGURE 2 Vertical shifts FIGURE 3 Horizontal shifts MATCHED PROBLEM 2 (A) How are the graphs of = 1 + and = 1-4 related to the graph of = 1? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. (B) How are the graphs of = 1 + and = 1-4 related to the graph of = 1? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. Comparing the graphs of = f() + k with the graph of = f(), we see that the graph of = f() + k can be obtained from the graph of = f() b verticall translating (shifting) the graph of the latter upward k units if k is positive and downward ƒ k ƒ units if k is negative. Comparing the graphs of = f( + h) with the graph of = f(), we see that the graph of = f( + h) can be obtained from the graph

22 Section 2.2 Elementar Functions: Graphs and Transformations 67 of = f() b horizontall translating (shifting) the graph of the latter h units to the left if h is positive and ƒ h ƒ units to the right if h is negative. EXAMPLE 3 Vertical and Horizontal Translations (Shifts) The graphs in Figure 4 are either horizontal or vertical shifts of the graph of f() = 2. Write appropriate equations for functions H, G, M, and N in terms of f. H f G M f N (A) (B) FIGURE 4 Vertical and horizontal shifts SOLUTION Functions H and G are vertical shifts given b H() = G() = 2-4 Functions M and N are horizontal shifts given b M() = ( + 2) 2 N() = ( - 3) 2 MATCHED PROBLEM 3 The graphs in Figure are either horizontal or vertical shifts of the graph of f() = 1. 3 Write appropriate equations for functions H, G, M, and N in terms of f. M f H f G N (A) (B) FIGURE Vertical and horizontal shifts Reflections, Stretches, and Shrinks We now investigate how the graph of = Af() is related to the graph of = f() for different real numbers A.

23 68 CHAPTER 2 Functions and Graphs Eplore & Discuss 2 1 (A) Graph = A 2 for A = 1, 4, and 4 simultaneousl in the same coordinate sstem. (B) Graph = A 2 for A = -1, -4, and simultaneousl in the same coordinate sstem. (C) Describe the relationship between the graph of h() = 2 and the graph of G() = A 2 for A an real number. Comparing = Af() to = f(), we see that the graph of = Af() can be obtained from the graph of = f() b multipling each ordinate value of the latter b A. The result is a vertical stretch of the graph of = f() if A 7 1, a vertical shrink of the graph of = f() if 0 6 A 6 1, and a reflection in the ais if A = -1. If A is a negative number other than -1, then the result is a combination of a reflection in the ais and either a vertical stretch or a vertical shrink. EXAMPLE 4 Reflections, Stretches, and Shrinks (A) How are the graphs of = 2 ƒ ƒ and = 0. ƒ ƒ related to the graph of = ƒ ƒ? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. (B) How is the graph of = -2 ƒ ƒ related to the graph of = ƒ ƒ? Confirm our answer b graphing both functions simultaneousl in the same coordinate sstem. SOLUTION (A) The graph of = 2 ƒ ƒ is a vertical stretch of the graph of = ƒ ƒ b a factor of 2, and the graph of = 0. ƒ ƒ is a vertical shrink of the graph of = ƒ ƒ b a factor of 0.. Figure 6 confirms this conclusion. (B) The graph of = -2 ƒ ƒ is a reflection in the ais and a vertical stretch of the graph of = ƒ ƒ. Figure 7 confirms this conclusion FIGURE 6 Vertical stretch and shrink FIGURE 7 Reflection and vertical stretch MATCHED PROBLEM 4 (A) How are the graphs of = 2 and = 0. related to the graph of =? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. (B) How is the graph of = -0. related to the graph of =? Confirm our answer b graphing both functions in the same coordinate sstem. The various transformations considered above are summarized in the following bo for eas reference:

24 Section 2.2 Elementar Functions: Graphs and Transformations 69 SUMMARY GRAPH TRANSFORMATIONS Vertical Translation: = f() + k e k 7 0 k 6 0 Horizontal Translation: = f( + h) e h 7 0 h 6 0 Shift graph of = f() up k units. Shift graph of = f() down ƒ k ƒ units. Shift graph of = f() left h units. Shift graph of = f() right ƒ h ƒ units. Reflection: = -f() Reflect the graph of = f() in the ais. Vertical Stretch and Shrink: = Af() µ A 7 1 Stretch graph of = f() verticall b multipling each ordinate value b A. 0 6 A 6 1 Shrink graph of = f() verticall b multipling each ordinate value b A. Eplore & Discuss 3 Use a graphing calculator to eplore the graph of = A( + h) 2 + k for various values of the constants A, h, and k. Discuss how the graph of = A( + h) 2 + k is related to the graph of = 2. EXAMPLE SOLUTION Combining Graph Transformations Discuss the relationship between the graphs of = -ƒ - 3 ƒ + 1 and = ƒ ƒ. Confirm our answer b graphing both functions simultaneousl in the same coordinate sstem. The graph of = -ƒ - 3 ƒ + 1 is a reflection in the ais, a horizontal translation of 3 units to the right, and a vertical translation of 1 unit upward of the graph of = ƒ ƒ. Figure 8 confirms this description. 3 1 FIGURE 8 Combined transformations MATCHED PROBLEM The graph of = G() in Figure 9 on the net page involves a reflection and a translation of the graph of = 3. Describe how the graph of function G is related to the graph of = 3 and find an equation of the function G.

25 70 CHAPTER 2 Functions and Graphs G FIGURE 9 Combined transformations Piecewise-Defined Functions Earlier we noted that the absolute value of a real number can be defined as - if 6 0 ƒ ƒ = e if Ú 0 Notice that this function is defined b different rules for different parts of its domain. Functions whose definitions involve more than one rule are called piecewise-defined functions. Graphing one of these functions involves graphing each rule over the appropriate portion of the domain (Fig. ). In Figure C, notice that an open dot is used to show that the point (0, -2) is not part of the graph and a solid dot is used to show that (0, 2) is part of the graph. As the net eample illustrates, piecewise-defined functions occur naturall in man applications. (A) 2 2 (B) 2 2 (C) 2 2 if if 0 FIGURE Graphing a piecewise-defined function EXAMPLE 6 Natural Gas Rates Easton Utilities uses the rates shown in Table 1 to compute the monthl cost of natural gas for each customer. Write a piecewise definition for the cost of consuming CCF (cubic hundred feet) of natural gas and graph the function. TABLE 1 Charges per Month $ per CCF for the first CCF $ per CCF for the net 3 CCF $0.208 per CCF for all over 40 CCF SOLUTION If C() is the cost, in dollars, of using CCF of natural gas in one month, then the first line of Table 1 implies that C() = if 0

26 Section 2.2 Elementar Functions: Graphs and Transformations 71 $30 $20 $ C() FIGURE 11 Cost of purchasing CCF of natural gas Note that C() = is the cost of CCF. If 6 40, then - represents the amount of gas that cost $ per CCF, ( - ) represents the cost of this gas, and the total cost is If 7 40, then C() = ( - ) C() = ( - 40) where = C(40), the cost of the first 40 CCF. Combining all these equations, we have the following piecewise definition for C(): if 0 C() = µ ( - ) if ( - 40) if 40 6 To graph C, first note that each rule in the definition of C represents a transformation of the identit function f() =. Graphing each transformation over the indicated interval produces the graph of C shown in Figure 11. MATCHED PROBLEM 6 Natural Gas Rates Trussville Utilities uses the rates shown in Table 2 to compute the monthl cost of natural gas for residential customers. Write a piecewise definition for the cost of consuming CCF of natural gas and graph the function. TABLE 2 Charges per Month $0.767 per CCF for the first 0 CCF $ per CCF for the net CCF $ per CCF for all over 200 CCF Answers to Matched Problems 1. (A) f(729) = 729, h(729) = 31,441, m(729) = 387,420,489, n(729) = 27, p(729) = 9, g(729) = 729 (B) f(-.2) = -.2, h(-.2) = 27.62, m(-.2) = , n(-.2) is not a real number, p(-.2) = , g(-.2) =.2 2. (A) The graph of = 1 + is the same as the graph of = 1 shifted upward units, and the graph of = 1-4 is the same as the graph of = 1 shifted downward 4 units. The figure confirms these conclusions. 4

27 72 CHAPTER 2 Functions and Graphs (B) The graph of = 1 + is the same as the graph of = 1 shifted to the left units, and the graph of = 1-4 is the same as the graph of = 1 shifted to the right 4 units. The figure confirms these conclusions H() = , G() = 1 3-2, M() = , N() = (A) The graph of = 2 is a vertical stretch of the graph of =, and the graph of = 0. is a vertical shrink of the graph of =. The figure confirms these conclusions (B) The graph of = -0. is a vertical shrink and a reflection in the ais of the graph of =. The figure confirms this conclusion. 0.. The graph of function G is a reflection in the ais and a horizontal translation of 2 units to the left of the graph of = 3. An equation for G is G() = -( + 2) if C() = µ ( - 0) if ( - 200) if C()

28 Section 2.2 Elementar Functions: Graphs and Transformations 73 Eercise 2-2 A Without looking back in the tet, indicate the domain and range of each of the functions in Problems f() = 2 2. g() = ƒ ƒ 3. h() = m() = r() = k() = 41 2 n() = -0.1 s() = 3 1 Graph each of the functions in Problems 9 20 using the graphs of functions f and g below. f() g() = f() + 2. = g() = f( + 2) 12. = g( - 1) 13. = g( - 3) = g() = f( + 3) = f() + 3 B 17. = -f() = 0.g() 20. = -g() = 2f() In Problems 21 28, indicate verball how the graph of each function is related to the graph of one of the si basic functions in Figure 1 on page 6. Sketch a graph of each function g() = -ƒ + 3 ƒ h() = -ƒ - ƒ f() = ( - 4) 2-3 m() = ( + 3) f() = 7-1 g() = -6 + h() = -3 ƒ ƒ m() = Each graph in Problems is the result of appling a sequence of transformations to the graph of one of the si basic functions in Figure 1 on page 6. Identif the basic function and describe the transformation verball. Write an equation for the given graph In Problems 37 42, the graph of the function g is formed b appling the indicated sequence of transformations to the given function f. Find an equation for the function g and graph g using - and The graph of f() = 1 is shifted 2 units to the right and 3 units down. 38. The graph of f() = 1 3 is shifted 3 units to the left and 2 units up. 39. The graph of f() = ƒ ƒ is reflected in the ais and shifted to the left 3 units. 40. The graph of f() = ƒ ƒ is reflected in the ais and shifted to the right 1 unit. 41. The graph of f() = 3 is reflected in the ais and shifted 2 units to the right and down 1 unit. 42. The graph of f() = 2 is reflected in the ais and shifted to the left 2 units and up 4 units.

29 74 CHAPTER 2 Functions and Graphs Graph each function in Problems if 6 2 f() = b - 2 if Ú if 6-1 g() = b if Ú if 0 h() = b if if 0 20 h() = b if if 0 20 h() = c + 20 if if if 0 20 h() = c if if 7 0 C Each of the graphs in Problems 49 4 involves a reflection in the ais and/or a vertical stretch or shrink of one of the basic functions in Figure 1 on page 6. Identif the basic function, and describe the transformation verball. Write an equation for the given graph Changing the order in a sequence of transformations ma change the final result. Investigate each pair of transformations in Problems 60 to determine if reversing their order can produce a different result. Support our conclusions with specific eamples and/or mathematical arguments.. Vertical shift; horizontal shift 6. Vertical shift; reflection in ais 7. Vertical shift; reflection in ais 8. Vertical shift; vertical stretch 9. Horizontal shift; reflection in ais 60. Horizontal shift; vertical shrink Applications 61. Price demand. A retail chain sells CD plaers. The retail price p() (in dollars) and the weekl demand for a particular model are related b p() = (A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure 1 on page 6. (B) Sketch a graph of function p using part (A) as an aid. 62. Price suppl. The manufacturers of the CD plaers in Problem 61 are willing to suppl plaers at a price of p() as given b the equation p() = (A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure 1 on page 6. (B) Sketch a graph of function p using part (A) as an aid. 63. Hospital costs. Using statistical methods, the financial department of a hospital arrived at the cost equation C() = ( - 00) , ,000 where C() is the cost in dollars for handling cases per month. (A) Describe how the graph of function C can be obtained from the graph of one of the basic functions in Figure 1 on page 6. (B) Sketch a graph of function C using part (A) and a graphing calculator as aids. 64. Price demand. A compan manufactures and sells in-line skates. Its financial department has established the price demand function p() = ( - ) 2 0

30 Section 2.2 Elementar Functions: Graphs and Transformations 7 where p() is the price at which thousand pairs of skates can be sold. (A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure 1 on page 6. (B) Sketch a graph of function p using part (A) and a graphing calculator as aids. 6. Electricit rates. Table 3 shows the electricit rates charged b Monroe Utilities in the summer months. The base is a fied monthl charge, independent of the kwh (kilowatthours) used during the month. (A) Write a piecewise definition of the monthl charge S() for a customer who uses kwh in a summer month. (B) Graph S(). TABLE 3 Summer (Jul October) Base charge, $8.0 First 700 kwh or less at 0.060/kWh Over 700 kwh at /kWh 66. Electricit rates. Table 4 shows the electricit rates charged b Monroe Utilities in the winter months. (A) Write a piecewise definition of the monthl charge W() for a customer who uses kwh in a winter month. TABLE 6 Kansas State Income Ta SCHEDULE II SINGLE, HEAD OF HOUSEHOLD, OR MARRIED FILING SEPARATE If taable income is Over But Not Over Ta Due Is $0 $1, % of taable income $1,000 $30,000 $2 plus 6.2% of ecess over $1,000 $30,000 $1,462.0 plus 6.4% of ecess over $30,000 (A) Write a piecewise definition for the ta due T() on an income of dollars. (B) Graph T(). (C) Find the ta due on a taable income of $20,000. Of $3,000. (D) Would it be better for a married couple in Kansas with two equal incomes to file jointl or separatel? Discuss. 69. Phsiolog. A good approimation of the normal weight of a person 60 inches or taller but not taller than 80 inches is given b w() = , where is height in inches and w() is weight in pounds. TABLE 4 Winter (November June) Base charge, $8.0 First 700 kwh or less at 0.060/kWh Over 700 kwh at 0.030/kWh TABLE (B) Graph W(). 67. State income ta. Table shows a recent state income ta schedule for married couples filing a joint return in the state of Kansas. (A) Write a piecewise definition for the ta due T() on an income of dollars. (B) Graph T(). (C) Find the ta due on a taable income of $40,000. Of $70,000. Kansas State Income Ta SCHEDULE I MARRIED FILING JOINT If taable income is Over But Not Over Ta Due Is $0 $30, % of taable income $30,000 $60,000 $1,00 plus 6.2% of ecess over $30,000 $60,000 $2,92 plus 6.4% of ecess over $60, State income ta. Table 6 shows a recent state income ta schedule for individuals filing a return in the state of Kansas. (A) Describe how the graph of function w can be obtained from the graph of one of the basic functions in Figure 1, page 6. (B) Sketch a graph of function w using part (A) as an aid. 70. Phsiolog. The average weight of a particular species of snake is given b w() = 463 3, , where is length in meters and w() is weight in grams. (A) Describe how the graph of function w can be obtained from the graph of one of the basic functions in Figure 1, page 6. (B) Sketch a graph of function w using part (A) as an aid. 71. Safet research. Under ideal conditions, if a person driving a vehicle slams on the brakes and skids to a stop, the speed of the vehicle v() (in miles per hour) is given approimatel b v() = C1, where is the length of skid marks (in feet) and C is a constant that depends on the road conditions and the weight of the vehicle. For a particular vehicle, v() = and (A) Describe how the graph of function v can be obtained from the graph of one of the basic functions in Figure 1, page 6. (B) Sketch a graph of function v using part (A) as an aid.

31 76 CHAPTER 2 Functions and Graphs 72. Learning. A production analst has found that on the average it takes a new person T() minutes to perform a particular assembl operation after performances of the operation, where T() = - 1, (A) Describe how the graph of function T can be obtained from the graph of one of the basic functions in Figure 1, page 6. (B) Sketch a graph of function T using part (A) as an aid. Section 2-3 h() FIGURE 1 Square function h() = 2 Eplore & Discuss 1 QUADRATIC FUNCTIONS Quadratic Functions, Equations, and Inequalities Properties of Quadratic Functions and Their Graphs Applications More General Functions: Polnomial and Rational Functions If the degree of a linear function is increased b one, we obtain a second-degree function, usuall called a quadratic function, another basic function that we will need in our librar of elementar functions. We will investigate relationships between quadratic functions and the solutions to quadratic equations and inequalities. Other important properties of quadratic functions will also be investigated, including maimum and minimum properties. We will then be in a position to solve important practical problems such as finding production levels that will produce maimum revenue or maimum profit. Quadratic Functions, Equations, and Inequalities The graph of the square function h() = 2 is shown in Figure 1. Notice that the graph is smmetric with respect to the ais and that (0, 0) is the lowest point on the graph. Let s eplore the effect of appling a sequence of basic transformations to the graph of h. Indicate how the graph of each function is related to the graph of the function h() = 2. Find the highest or lowest point, whichever eists, on each graph. (A) f() = ( - 3) 2-7 = (B) g() = 0.( + 2) = (C) m() = -( - 4) = (D) n() = -3( + 1) 2-1 = Graphing the functions in Eplore Discuss 1 produces figures similar in shape to the graph of the square function in Figure 1. These figures are called parabolas. The functions that produced these parabolas are eamples of the important class of quadratic functions, which we now define. DEFINITION Quadratic Functions If a, b, and c are real numbers with a Z 0, then the function f() = a 2 + b + c Standard form is a quadratic function and its graph is a parabola. INSIGHT If is an real number, then a 2 + b + c is also a real number. According to the agreement on domain and range in Section 2-1, the domain of a quadratic function is R, the set of real numbers. We will discuss methods for determining the range of a quadratic function later in this section. Tpical graphs of quadratic functions are illustrated in Figure 2.

32 Section 2.3 Quadratic Functions 77 f() g() h() (A) f() 2 4 (B) g() (C) h() INSIGHT FIGURE 2 Graphs of quadratic functions The intercepts of a linear function can be found b solving the linear equation = m + b = 0 for, m Z 0 (see Section 1-2). Similarl, the intercepts of a quadratic function can be found b solving the quadratic equation = a 2 + b + c = 0 for, a Z 0. Several methods for solving quadratic equations are discussed in Appendi A, Section A-7. The most popular of these is the quadratic formula. If a 2 + b + c = 0, a Z 0, then = -b ; 2b2-4ac, provided b 2-4ac Ú 0 2a EXAMPLE 1 Intercepts, Equations, and Inequalities (A) Sketch a graph of f() = in a rectangular coordinate sstem. (B) Find and intercepts algebraicall to four decimal places. (C) Graph f() = in a standard viewing window. (D) Find the and intercepts to four decimal places using TRACE and zero on our graphing calculator. (E) Solve the quadratic inequalit Ú 0 graphicall to four decimal places using the results of parts (A) and (B) or (C) and (D). (F) Solve the equation = 4 graphicall to four decimal places using INTERSECT on our graphing calculator. SOLUTION (A) Hand-sketching a graph of f: f() (B) Finding intercepts algebraicall: intercept: f(0) = -(0) 2 + (0) + 3 = 3 intercepts: f() = = 0 Quadratic equation

33 78 CHAPTER 2 Functions and Graphs = -b ; 2b2-4ac 2a = -() ; 22-4(-1)(3) 2(-1) - ; 237 = = or (C) Graphing in a graphing calculator: Quadratic formula (see Appendi A-7) (D) Finding intercepts graphicall using a graphing calculator: intercept: intercept: intercept: 3 (E) Solving Ú 0 graphicall: The quadratic inequalit Ú 0 holds for those values of for which the graph of f() = in the figures in parts (A) and (C) is at or above the ais. This happens for between the two intercepts [found in part (B) or (D)], including the two intercepts. Thus, the solution set for the quadratic inequalit is or [ ,.414]. (F) Solving the equation = 4 using a graphing calculator: = 4 at = = 4 at = MATCHED PROBLEM 1 (A) Sketch a graph of g() = in a rectangular coordinate sstem. (B) Find and intercepts algebraicall to four decimal places. (C) Graph g() = in a standard viewing window. (D) Find the and intercepts to four decimal places using TRACE and the zero command on our graphing calculator. (E) Solve Ú 0 graphicall to four decimal places using the results of parts (A) and (B) or (C) and (D). (F) Solve the equation = -3 graphicall to four decimal places using an appropriate built-in routine in our graphing calculator.

34 Section 2.3 Quadratic Functions 79 Eplore & Discuss 2 How man intercepts can the graph of a quadratic function have? How man intercepts? Eplain our reasoning. f() Maimum: f(4) 8 f() ( 4) 2 8 Line of smmetr: 4 FIGURE 3 Graph of a quadratic function Properties of Quadratic Functions and Their Graphs Man useful properties of the quadratic function can be uncovered b transforming into the verte form* The process of completing the square (see Appendi A-7) is central to the transformation. We illustrate the process through a specific eample and then generalize the results. Consider the quadratic function given b f() = (1) We use completing the square to transform this function into verte form: Thus, f() = = -2( 2-8) - 24 = -2( 2-8 +?) - 24 = -2( ) = -2( - 4) f() = a 2 + b + c a Z 0 f() = a( - h) 2 + k f() = -2( - 4) If = 4, then -2( - 4) 2 = 0 and f(4) = 8. For an other value of, the negative number -2( - 4) 2 is added to 8, making it smaller. (Think about this.) Therefore, f(4) = 8 is the maimum value of f() for all a ver important result! Furthermore, if we choose an two values that are the same distance from 4, we will obtain the same function value. For eample, = 3 and = are each one unit from = 4 and their function values are f(3) = -2(3-4) = 6 f() = -2( - 4) = 6 Factor the coefficient of the first two terms. out of Add 16 to complete the square inside the parentheses. Because of the -2 outside the parentheses, we have actuall added -32, so we must add 32 to the outside. The transformation is complete and can be checked b multipling out. Thus, the vertical line = 4 is a line of smmetr.that is, if the graph of equation (1) is drawn on a piece of paper and the paper is folded along the line = 4, then the two sides of the parabola will match eactl. All these results are illustrated b graphing equations (1) and (2) and the line = 4 simultaneousl in the same coordinate sstem (Fig. 3). From the preceding discussion, we see that as moves from left to right, f() is increasing on (-q, 4], and decreasing on [4, q), and that f() can assume no value greater than 8. Thus, Range of f : 8 or (- q, 8] 2 (2) * This terminolog is not universall agreed upon. Some call this the standard form.

35 80 CHAPTER 2 Functions and Graphs In general, the graph of a quadratic function is a parabola with line of smmetr parallel to the vertical ais. The lowest or highest point on the parabola, whichever eists, is called the verte. The maimum or minimum value of a quadratic function alwas occurs at the verte of the parabola.the line of smmetr through the verte is called the ais of the parabola. In the eample above, = 4 is the ais of the parabola and (4, 8) is its verte. INSIGHT Appling the graph transformation properties discussed in Section 2-2 to the transformed equation, f() = = -2( - 4) we see that the graph of f() = is the graph of g() = 2 verticall stretched b a factor of 2, reflected in the ais, and shifted to the right 4 units and up 8 units, as shown in Figure 4. g() 2 f() ( 4) 2 8 FIGURE 4 Graph of f is the graph of g transformed Note the important results we have obtained from the verte form of the quadratic function f: The verte of the parabola The ais of the parabola The maimum value of f() The range of the function f The relationship between the graph of g() = 2 and the graph of f() = Now, let us eplore the effects of changing the constants a, h, and k on the graph of f() = a( - h) 2 + k. Eplore & Discuss 3 (A) Let a = 1 and h =. Graph f() = a( - h) 2 + k for k = -4, 0, and 3 simultaneousl in the same coordinate sstem. Eplain the effect of changing k on the graph of f. (B) Let a = 1 and k = 2. Graph f() = a( - h) 2 + k for h = -4, 0, and simultaneousl in the same coordinate sstem. Eplain the effect of changing h on the graph of f. (C) Let h = and k = -2. Graph f() = a( - h) 2 + k for a = 0.2, 1, and 3 simultaneousl in the same coordinate sstem. Graph function f for

36 Section 2.3 Quadratic Functions 81 a = 1, -1, and -0.2 simultaneousl in the same coordinate sstem. Eplain the effect of changing a on the graph of f. (D) Discuss parts (A) (C) using a graphing calculator and a standard viewing window. The preceding discussion is generalized for all quadratic functions in the following summar: SUMMARY PROPERTIES OF A QUADRATIC FUNCTION AND ITS GRAPH Given a quadratic function and the verte form obtained b completing the square = a( - h) 2 + k we summarize general properties as follows: 1. The graph of f is a parabola: f() = a 2 + b + c a Z 0 Standard form Verte form f() Ais h f() Ais h k h Verte (h, k) Min f() k h Verte (h, k) Ma f() a 0 Opens upward a 0 Opens downward 2. Verte: (h, k) (parabola increases on one side of the verte and decreases on the other) 3. Ais (of smmetr): = h (parallel to ais) 4. f(h) = k is the minimum if a 7 0 and the maimum if a 6 0. Domain: All real numbers Range: (- q, k] if a 6 0 or [k, q) if a The graph of f is the graph of g() = a 2 translated horizontall h units and verticall k units. EXAMPLE 2 Analzing a Quadratic Function Given the quadratic function f() = (A) Find the verte form for f. (B) Find the verte and the maimum or minimum. State the range of f. (C) Describe how the graph of function f can be obtained from the graph of g() = 2 using transformations discussed in Section 2-2. (D) Sketch a graph of function f in a rectangular coordinate sstem. (E) Graph function f using a suitable viewing window. (F) Find the verte and the maimum or minimum graphicall using the minimum command. State the range of f.

37 82 CHAPTER 2 Functions and Graphs SOLUTION (A) Complete the square to find the verte form: f() = = 0.( ?) + 21 = 0.( ) = 0.( - 6) (B) From the verte form, we see that h = 6 and k = 3. Thus, verte: (6, 3); minimum: f(6) = 3; range: Ú 3 or [3, q). (C) The graph of f() = 0.( - 6) is the same as the graph of g() = 2 verticall shrunk b a factor of 0., and shifted to the right 6 units and up 3 units. (D) Graph in a rectangular coordinate sstem: f() (E) Graph in a graphing calculator: (F) Finding the verte, minimum, and range graphicall using a graphing calculator: Verte: (6, 3); minimum: f(6) = 3; range: Ú 3 or [3, q). MATCHED PROBLEM 2 Given the quadratic function f() = (A) Find the verte form for f. (B) Find the verte and the maimum or minimum. State the range of f. (C) Describe how the graph of function f can be obtained from the graph of g() = 2 using transformations discussed in Section 2-2. (D) Sketch a graph of function f in a rectangular coordinate sstem. (E) Graph function f using a suitable viewing window. (F) Find the verte and the maimum or minimum graphicall using the minimum command. State the range of f.

38 Section 2.3 Quadratic Functions 83 APPLICATIONS EXAMPLE 3 SOLUTION Maimum Revenue This is a continuation of Eample 6 in Section 2-1. Recall that the financial department in the compan that produces a digital camera arrived at the following price demand function and the corresponding revenue function: p() = Price demand function R() = p() = ( ) Revenue function where p() is the wholesale price per camera at which million cameras can be sold and R() is the corresponding revenue (in million dollars). Both functions have domain 1 1. (A) Find the output to the nearest thousand cameras that will produce the maimum revenue. What is the maimum revenue to the nearest thousand dollars? Solve the problem algebraicall b completing the square. (B) What is the wholesale price per camera (to the nearest dollar) that produces the maimum revenue? (C) Graph the revenue function using an appropriate viewing window. (D) Find the output to the nearest thousand cameras that will produce the maimum revenue. What is the maimum revenue to the nearest thousand dollars? Solve the problem graphicall using the maimum command. (A) Algebraic solution: The maimum revenue of million dollars ($449,32,000) occurs when = million cameras (9,480,000 cameras). (B) Finding the wholesale price per camera: Use the price demand function for an output of million cameras: (C) Graph in a graphing calculator: 00 R() = ( ) = = -( ?) = -( ) = -( ) p() = p(9.480) = (9.480) = $ (D) Graphical solution using a graphing calculator: An output of million cameras (9,480,000 cameras) will produce a maimum revenue of million dollars ($449,32,000).

39 84 CHAPTER 2 Functions and Graphs MATCHED PROBLEM 3 The financial department in Eample 3, using statistical and analtical techniques (see Matched Problem 7 in Section 2-1), arrived at the cost function C() = Cost function where C() is the cost (in million dollars) for manufacturing and selling million cameras. (A) Using the revenue function from Eample 3 and the preceding cost function, write an equation for the profit function. (B) Find the output to the nearest thousand cameras that will produce the maimum profit. What is the maimum profit to the nearest thousand dollars? Solve the problem algebraicall b completing the square. (C) What is the wholesale price per camera (to the nearest dollar) that produces the maimum profit? (D) Graph the profit function using an appropriate viewing window. (E) Find the output to the nearest thousand cameras that will produce the maimum profit. What is the maimum profit to the nearest thousand dollars? Solve the problem graphicall using the maimum routine. EXAMPLE 4 Break-Even Analsis Use the revenue function from Eample 3 and the cost function from Matched Problem 3: R() = ( ) C() = Revenue function Cost function Both have domain 1 1. (A) Sketch the graphs of both functions in the same coordinate sstem. (B) Break-even points are the production levels at which R() = C(). Find the break-even points algebraicall to the nearest thousand cameras. (C) Plot both functions simultaneousl in the same viewing window. (D) Use INTERSECT to find the break-even points graphicall to the nearest thousand cameras. (E) Recall that a loss occurs if R() 6 C() and a profit occurs if R() 7 C(). For what outputs (to the nearest thousand cameras) will a loss occur? A profit? SOLUTION (A) Sketch of functions: R() C() 00 Cost 20 Revenue 1

40 Section 2.3 Quadratic Functions 8 (B) Algebraic solution: Find such that R() = C(): ( ) = = 0 = -7.1 ; (-)(-16) 2(-) = -7.1 ; 22, = or The compan breaks even at = and million cameras. (C) Graph in a graphing calculator: Revenue 00 Cost 1 1 (D) Graphical solution: 0 Revenue 00 Cost Revenue 00 Cost The compan breaks even at = and million cameras. (E) Use the results from parts (A) and (B) or (C) and (D): Loss: or Profit: MATCHED PROBLEM 4 Use the profit equation from Matched Problem 3: P() = R() - C() = Profit function Domain: 1 1 (A) Sketch a graph of the profit function in a rectangular coordinate sstem. (B) Break-even points occur when P() = 0. Find the break-even points algebraicall to the nearest thousand cameras. (C) Plot the profit function in an appropriate viewing window. (D) Find the break-even points graphicall to the nearest thousand cameras. (E) A loss occurs if P() 6 0, and a profit occurs if P() 7 0. For what outputs (to the nearest thousand cameras) will a loss occur? A profit?

41 86 CHAPTER 2 Functions and Graphs A visual inspection of the plot of a data set might indicate that a parabola would be a better model of the data than a straight line. In that case, rather than using linear regression (Section 1-3) to fit a linear model to the data, we would use quadratic regression on a graphing calculator to find the function of the form = a 2 + b + c that best fits the data. EXAMPLE Outboard Motors Table 1 gives performance data for a boat powered b an Evinrude outboard motor. Use quadratic regression to find the best model of the form = a 2 + b + c for fuel consumption (in miles per gallon) as a function of speed (in miles per hour). Estimate the fuel consumption (to one decimal place) at a speed of 12 miles per hour. TABLE 1 rpm mph mpg 2, , , , , , , SOLUTION Enter the data in a graphing calculator (Fig. A) and find the quadratic regression equation (Fig. B). The data set and the regression equation are graphed in Figure C. Using TRACE we see that the estimated fuel consumption at a speed of 12 mph is 4. mpg. 0 0 (A) (B) FIGURE 0 (C) MATCHED PROBLEM Refer to Table 1. Use quadratic regression to find the best model of the form = a 2 + b + c for boat speed (in miles per hour) as a function of engine speed (in revolutions per minute). Estimate the boat speed (in miles per hour, to one decimal place) at an engine speed of 3,400 rpm. More General Functions: Polnomial and Rational Functions Linear and quadratic functions are special cases of the more general class of polnomial functions. DEFINITION Polnomial Function A polnomial function is a function that can be written in the form f() = a n n + a n - 1 n Á + a 1 + a 0

42 Section 2.3 Quadratic Functions 87 for n a nonnegative integer, called the degree of the polnomial. The coefficients a 0, a 1,..., a n are real numbers with a n Z 0. The domain of a polnomial function is the set of all real numbers. Figure 6 shows graphs of representative polnomial functions of degrees 1 through 6. The figure suggests some general properties of graphs of polnomial functions. f() g() h() (A) f() 2 (B) g() 3 2 (C) h() F() G() H() (D) F() (E) G() (F) H() FIGURE 6 Graphs of polnomial functions Notice that the odd-degree polnomial graphs start negative, end positive, and cross the ais at least once. The even-degree polnomial graphs start positive, end positive, and ma not cross the ais at all. In all cases in Figure 1, the coefficient of the highest-degree term was chosen positive. If an leading coefficient had been chosen negative, then we would have a similar graph but reflected in the ais. A polnomial of degree n can have at most n linear factors. Therefore, the graph of a polnomial function of positive degree n can intersect the ais at most n times. Note from Figure 6 that a polnomial of degree n ma intersect the ais fewer than n times. The graph of a polnomial function is continuous, with no holes or breaks.that is, the graph can be drawn without removing a pen from the paper.also, the graph of a polnomial has no sharp corners. Figure 7 on the net page shows the graphs of two functions, one that is not continuous, and the other that is continuous, but with a sharp corner. Neither function is a polnomial. Just as rational numbers are defined in terms of quotients of integers, rational functions are defined in terms of quotients of polnomials. The following equations specif rational functions: f() = 1 g() = p() = q() = 7 r() = 0 h() = 3-8

43 88 CHAPTER 2 Functions and Graphs f() h() Discontinuous break at 0 2 (A) f() (B) h() 3 Continuous, but sharp corner at (0, 3) FIGURE 7 Discontinuous and sharp-corner functions DEFINITION Rational Function A rational function is an function that can be written in the form f() = n() d() d() Z 0 where n() and d() are polnomials. The domain is the set of all real numbers such that d() Z 0. Figure 8 shows the graphs of representative rational functions. Note, for eample, that in Figure 8A the line = 2 is a vertical asmptote for the function.the graph of f gets closer to this line as gets closer to 2.The line = 1 in Figure 8A is a horizontal asmptote for the function. The graph of f gets closer to this line as increases or decreases without bound. f() f() f() (A) f() 3 2 (B) f() (C) f() 1 FIGURE 8 Graphs of rational functions The number of vertical asmptotes of a rational function f() = n()>d() is at most equal to the degree of d(). A rational function has at most one horizontal asmptote (note that the graph in Fig. 8C does not have a horizontal asmptote). Moreover, the graph of a rational function approaches the horizontal asmptote (when one eists) both as increases and decreases without bound. Answers to Matched Problems 1. (A) (B) intercepts: , 3.266; intercept: - g() (C)

44 (D) intercepts: , 3.266; intercept: (E) or Ú 3.266; or (-q, or , q) (F) = , Section 2.3 Quadratic Functions (A) f() = -0.2( + 4) (B) Verte: (-4, 6) ; maimum: f(-4) = 6; range: 6 or (- q, 6] (C) The graph of f() = -0.2( + 4) is the same as the graph of g() = 2 verticall shrunk b a factor of 0.2, reflected in the ais, and shifted 4 units to the left and 6 units up. (D) (E) (F) Verte: (-4, 6) ; maimum: f(-4) = 6; range: 6 or (- q, 6] 3. (A) P() = R() - C() = (B) P() = R() - C() = -( - 7.1) ; output of 7. million cameras will produce a maimum profit of million dollars. (C) p(7.) = $7 (D) (E) An output of 7.1 million cameras will produce a maimum profit of million dollars. (Notice that maimum profit does not occur at the same output where maimum revenue occurs.) 4. (A) P() (B) = or million cameras (C)

45 90 CHAPTER 2 Functions and Graphs (D) = or million cameras (E) Loss: or ; profit: mph Eercise 2-3 A In Problems 1 4, complete the square and find the standard form of each quadratic function. 1. f() = g() = m() = n() = In Problems 8, write a brief verbal description of the relationship between the graph of the indicated function (from Problems 1 4) and the graph of = 2.. f() = g() = m() = n() = Match each equation with a graph of one of the functions f, g, m, or n in the figure. (A) = -( + 2) (B) = ( - 2) 2-1 (C) = ( + 2) 2-1 (D) = -( - 2) f g For the functions indicated in Problems 11 14, find each of the following to the nearest integer b referring to the graphs for Problems 9 and. (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range (E) Increasing interval (F) Decreasing interval 11. Function n in the figure for Problem Function m in the figure for Problem 13. Function f in the figure for Problem Function g in the figure for Problem In Problems 1 18, find each of the following: (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range 1. f() = -( - 3) g() = -( + 2) m() = ( + 1) n() = ( - 4) 2-3 B In Problems 19 22, write an equation for each graph in the form = a( - h) 2 + k, where a is either 1 or -1 and h and k are integers. m Figure for 9 n Match each equation with a graph of one of the functions f, g, m, or n in the figure. (A) = ( - 3) 2-4 (B) = -( + 3) (C) = -( - 3) (D) = ( + 3) 2-4 f g m n Figure for

46 Section 2.3 Quadratic Functions 91 C In Problems 23 28, find the verte form for each quadratic function. Then find each of the following: (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range 23. f() = g() = r() = s() = u() = v() = Let f() = Solve each equation graphicall to four decimal places. (A) f() = 4 (B) f() = -1 (C) f() = Let g() = Solve each equation graphicall to four decimal places. (A) g() = -2 (B) g() = (C) g() = Let f() = Find the maimum value of f to four decimal places graphicall. 32. Let f() = Find the maimum value of f to four decimal places graphicall. 33. Eplain under what graphical conditions a quadratic function has eactl one real zero. 34. Eplain under what graphical conditions a quadratic function has no real zeros. In Problems 3 38, first write each function in verte form; then find each of the following (to four decimal places): (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range 3. g() = m() = f() = n() = Solve Problems graphicall to two decimal places using a graphing calculator = 0 = Ú Given that f is a quadratic function with minimum f() = f(2) = 4, find the ais, verte, range, and intercepts. 46. Given that f is a quadratic function with maimum f() = f(-3) = -, find the ais, verte, range, and intercepts. In Problems 47 0, (A) Graph f and g in the same coordinate sstem. (B) Solve f() = g() algebraicall to two decimal places. (C) Solve f() 7 g() using parts (A) and (B). (D) Solve f() 6 g() using parts (A) and (B) f() = -0.4( - ) g() = f() = -0.7( - 7) g() = f() = g() = f() = g() = Give a simple eample of a quadratic function that has no real zeros. Eplain how its graph is related to the ais. 2. Give a simple eample of a quadratic function that has eactl one real zero. Eplain how its graph is related to the ais. Applications 3. Tire mileage. An automobile tire manufacturer collected the data in the table relating tire pressure (in pounds per square inch) and mileage (in thousands of miles): Mileage A mathematical model for the data is given b f() = (A) Complete the following table. Round values of f() to one decimal place. Mileage f()

47 92 CHAPTER 2 Functions and Graphs (B) Sketch the graph of f and the mileage data in the same coordinate sstem. (C) Use values of the modeling function rounded to two decimal places to estimate the mileage for a tire pressure of 31 pounds per square inch. For 3 pounds per square inch. (D) Write a brief description of the relationship between tire pressure and mileage. 4. Automobile production. The table shows the retail market share of passenger cars from Ford Motor Compan as a percentage of the U.S. market. Year Market Share % % % % % % 7. Revenue. The marketing research department for a compan that manufactures and sells memor chips for microcomputers established the following price demand and revenue functions: p() = 7-3 R() = p() = (7-3) Price demand function Revenue function where p() is the wholesale price in dollars at which million chips can be sold, and R() is in millions of dollars. Both functions have domain (A) Sketch a graph of the revenue function in a rectangular coordinate sstem. (B) Find the output that will produce the maimum revenue. What is the maimum revenue? (C) What is the wholesale price per chip (to the nearest dollar) that produces the maimum revenue? A mathematical model for this data is given b f() = where = 0 corresponds to (A) Complete the following table. Market Share f() (B) Sketch the graph of f and the market share data in the same coordinate sstem. (C) Use values of the modeling function f to estimate Ford s market share in 20. In 201. (D) Write a brief verbal description of Ford s market share from 1980 to Tire mileage. Using quadratic regression on a graphing calculator, show that the quadratic function that best fits the data on tire mileage in Problem 3 is f() = (coefficients rounded to three significant digits). 6. Automobile production. Using quadratic regression on a graphing calculator, show that the quadratic function that best fits the data on market share in Problem 4 is f() = (coefficients rounded to three significant digits). 8. Revenue. The marketing research department for a compan that manufactures and sells notebook computers established the following price demand and revenue functions: p() = 2, R() = p() = (2,000-60) Price demand function Revenue function where p() is the wholesale price in dollars at which thousand computers can be sold, and R() is in thousands of dollars. Both functions have domain 1 2. (A) Sketch a graph of the revenue function in a rectangular coordinate sstem. (B) Find the output (to the nearest hundred computers) that will produce the maimum revenue. What is the maimum revenue to the nearest thousand dollars? (C) What is the wholesale price per computer (to the nearest dollar) that produces the maimum revenue? 9. Break-even analsis. Use the revenue function from Problem 7 in this eercise and the given cost function: R() = (7-3) C() = Revenue function Cost function where is in millions of chips, and R() and C() are in millions of dollars. Both functions have domain (A) Sketch a graph of both functions in the same rectangular coordinate sstem. (B) Find the break-even points to the nearest thousand chips. (C) For what outputs will a loss occur? A profit? 60. Break-even analsis. Use the revenue function from Problem 8, in this eercise and the given cost function: R() = (2,000-60) C() = 4, Revenue function Cost function where is thousands of computers, and C() and R() are in thousands of dollars. Both functions have domain 1 2.

48 Section 2.4 Eponential Functions 93 (A) Sketch a graph of both functions in the same rectangular coordinate sstem. (B) Find the break-even points. (C) For what outputs will a loss occur? Will a profit occur? 61. Profit loss analsis. Use the revenue and cost functions from Problem 9 in this eercise: R() = (7-3) C() = Revenue function Cost function where is in millions of chips, and R() and C() are in millions of dollars. Both functions have domain (A) Form a profit function P, and graph R, C, and P in the same rectangular coordinate sstem. (B) Discuss the relationship between the intersection points of the graphs of R and C and the intercepts of P. (C) Find the intercepts of P to the nearest thousand chips. Find the break-even points to the nearest thousand chips. (D) Refer to the graph drawn in part (A). Does the maimum profit appear to occur at the same output level as the maimum revenue? Are the maimum profit and the maimum revenue equal? Eplain. (E) Verif our conclusion in part (D) b finding the output (to the nearest thousand chips) that produces the maimum profit. Find the maimum profit (to the nearest thousand dollars), and compare with Problem 7B. 62. Profit loss analsis. Use the revenue and cost functions from Problem 60 in this eercise: R() = (2,000-60) C() = 4, Revenue function Cost function where is thousands of computers, and R() and C() are in thousands of dollars. Both functions have domain 1 2. (A) Form a profit function P, and graph R, C, and P in the same rectangular coordinate sstem. (B) Discuss the relationship between the intersection points of the graphs of R and C and the intercepts of P. (C) Find the intercepts of P to the nearest hundred computers. Find the break-even points. (D) Refer to the graph drawn in part (A). Does the maimum profit appear to occur at the same output level as the maimum revenue? Are the maimum profit and the maimum revenue equal? Eplain. (E) Verif our conclusion in part (D) b finding the output that produces the maimum profit. Find the maimum profit and compare with Problem 8B. 63. Medicine. The French phsician Poiseuille was the first to discover that blood flows faster near the center of an arter than near the edge. Eperimental evidence has shown that the rate of flow v (in centimeters per second) at a point centimeters from the center of an arter (see the figure) is given b v = f() = 1,000( ) Find the distance from the center that the rate of flow is 20 centimeters per second. Round answer to two decimal places. Arter Figure for 63 and Medicine. Refer to Problem 63. Find the distance from the center that the rate of flow is 30 centimeters per second. Round answer to two decimal places. 6. The table gives performance data for a boat powered b an Evinrude outboard motor. Find a quadratic regression model ( = a 2 + b + c) for boat speed (in miles per hour) as a function of engine speed (in revolutions per minute). Estimate the boat speed at 3,0 revolutions per minute. Table for 6 and 66 rpm mph mpg 1, , , , , The table gives performance data for a boat powered b an Evinrude outboard motor. Find a quadratic regression model ( = a 2 + b + c) for fuel consumption (in miles per gallon) as a function of engine speed (in revolutions per minute). Estimate the fuel consumption at 2,300 revolutions per minute. v Section 2-4 EXPONENTIAL FUNCTIONS Eponential Functions Base e Eponential Functions Growth and Deca Applications Compound Interest This section introduces the important class of functions called eponential functions. These functions are used etensivel in modeling and solving a wide variet of realworld problems, including growth of mone at compound interest; growth of populations of people, animals, and bacteria; radioactive deca; and learning associated with the master of such devices as a new computer or an assembl process in a manufacturing plant.

49 94 CHAPTER 2 Functions and Graphs Eponential Functions We start b noting that f() = 2 and g() = 2 are not the same function.whether a variable appears as an eponent with a constant base or as a base with a constant eponent makes a big difference. The function g is a quadratic function, which we have alread discussed. The function f is a new tpe of function called an eponential function. In general, DEFINITION Eponential Function The equation f() = b b 7 0, b Z 1 defines an eponential function for each different constant b, called the base. The domain of f is the set of all real numbers, and the range of f is the set of all positive real numbers. FIGURE 1 = 2 We require the base b to be positive to avoid imaginar numbers such as (-2) 1>2 = 2-2 = i22. We eclude b = 1 as a base, since f() = 1 = 1 is a constant function, which we have alread considered. Asked to hand-sketch graphs of equations such as = 2 or = 2 -, man students would not hesitate at all. 3Note: 2 - = 1>2 = (1>2).4 The would probabl make up tables b assigning integers to, plot the resulting points, and then join these points with a smooth curve as in Figure 1.The onl catch is that we have not defined 2 for all real numbers. From Appendi A, Section A-7, we know what 2, 2-3, 2 2>3, 2-3>, 2 1.4, and mean (that is, 2 p, where p is a rational number), but what does 2 22 mean? The question is not eas to answer at this time. In fact, a precise definition of 2 22 must wait for more advanced courses, where it is shown that 2 names a positive real number for an real number, and that the graph of = 2 is indeed as indicated in Figure 1. It is useful to compare the graphs of = 2 and = 2 - b plotting both on the same set of coordinate aes, as shown in Figure 2A. The graph of f() = b b 7 1 (Fig. 2B) (q) 2 2 b 0 b 1 b b 1 Domain (, ) Range (0, ) (A) FIGURE 2 Eponential functions (B)

50 Section 2.4 Eponential Functions 9 looks ver much like the graph of = 2, and the graph of f() = b 0 6 b 6 1 (Fig. 2B) looks ver much like the graph of = 2 -. Note that in both cases the ais is a horizontal asmptote for the graphs. The graphs in Figure 2 suggest the following important general properties of eponential functions, which we state without proof: THEOREM 1 BASIC PROPERTIES OF THE GRAPH OF f() b, b>0, b 1 1. All graphs will pass through the point (0, 1). b o = 1 for an permissible base b. 2. All graphs are continuous curves, with no holes or jumps. 3. The ais is a horizontal asmptote. 4. If b 7 1, then b increases as increases.. If 0 6 b 6 1, then decreases as increases. b INSIGHT Recall that the graph of a rational function has at most one horizontal asmptote and that it approaches the horizontal asmptote (if one eists) both as : q and as : - q (see Section 2-3). The graph of an eponential function, on the other hand, approaches its horizontal asmptote as : q or as : -q, but not both. In particular, there is no rational function that has the same graph as an eponential function. The use of a calculator with the ke, or its equivalent, makes the graphing of eponential functions almost routine. Eample 1 illustrates the process. EXAMPLE 1 Graphing Eponential Functions Sketch a graph of = ( 1 2 )4, SOLUTION Use a calculator to create the table of values shown. Plot these points, and then join them with a smooth curve as in Figure FIGURE 3 Graph of = ( 1 2 )4 MATCHED PROBLEM 1 Sketch a graph of = ( 1 2 )4 -, -2 2.

51 96 CHAPTER 2 Functions and Graphs Eplore & Discuss 1 Graph the functions f() = 2 and g() = 3 on the same set of coordinate aes. At which values of do the graphs intersect? For which values of is the graph of f above the graph of g? Below the graph of g? Are the graphs close together as increases without bound? Are the graphs close together as decreases without bound? Discuss. Eponential functions, whose domains include irrational numbers, obe the familiar laws of eponents discussed in Appendi A, Section A-6 for rational eponents. We summarize these eponent laws here and add two other important and useful properties. THEOREM 2 PROPERTIES OF EXPONENTIAL FUNCTIONS For a and b positive, a Z 1, b Z 1, and and real, 1. Eponent laws: a a = a + a a = a - (a ) = a (ab) = a b a a b b = a b = 4 = a = a if and onl if 3. For Z 0, a = b = If 7 t + 1 = 7 3t - 3, then t + 1 = 3t - 3, and t = -2. if and onl if a = b If a = 2, then a = 2. Base e Eponential Functions Of all the possible bases b we can use for the eponential function = b, which ones are the most useful? If ou look at the kes on a calculator, ou will probabl see and e. It is clear wh base would be important, because our number sstem is a base sstem. But what is e, and wh is it included as a base? It turns out that base e is used more frequentl than all other bases combined. The reason for this is that certain formulas and the results of certain processes found in calculus and more advanced mathematics take on their simplest form if this base is used. This is wh ou will see e used etensivel in epressions and formulas that model real-world phenomena. In fact, its use is so prevalent that ou will often hear people refer to = e as the eponential function. The base e is an irrational number, and like, it cannot be represented eactl b an finite decimal fraction. However, e can be approimated as closel as we like b evaluating the epression a1 + 1 b for sufficientl large. What happens to the value of epression (1) as increases without bound? Think about this for a moment before proceeding. Mabe ou guessed that the value approaches 1, because (1) approaches 1, and 1 raised to an power is 1. Let us see if this reasoning is correct b actuall calculating the value of the epression for larger and larger values of. Table 1 summarizes the results.

52 Section 2.4 Eponential Functions 97 TABLE 1 a1 + 1 b , , , ,000, Interestingl, the value of epression (1) is never close to 1, but seems to be approaching a number close to In fact, as increases without bound, the value of epression (1) approaches an irrational number that we call e. The irrational number e to 12 decimal places is e Compare this value of e with the value of from a calculator. Eactl who discovered the constant e is still being debated. It is named after the great Swiss mathematician Leonhard Euler ( ). e 1 DEFINITION Eponential Function with Base e Eponential functions with base e and base 1>e, respectivel, are defined b = e and = e - Domain: Range: 1- q, q) (0, q) e e Eplore & Discuss 2 Graph the functions f() = e, g() = 2, and h() = 3 on the same set of coordinate aes. At which values of do the graphs intersect? For positive values of, which of the three graphs lies above the other two? Below the other two? How does our answer change for negative values of? Growth and Deca Applications Functions of the form = ce kt, where c and k are constants and the independent variable t represents time, are often used to model population growth and radioactive deca. Note that if t = 0, then = c. So the constant c represents the initial population (or initial amount). The constant k is called the relative growth rate and

53 98 CHAPTER 2 Functions and Graphs has the following interpretation: Suppose that = ce kt models the population growth of a countr, where is the number of persons and t is time in ears. If the relative growth rate is k = 0.02, then at an time t, the population is growing at a rate of 0.02 persons (that is, 2% of the population) per ear. We sa that population is growing continuousl at relative growth rate k to mean that the population is given b the model = ce kt. EXAMPLE 2 Eponential Growth Cholera, an intestinal disease, is caused b a cholera bacterium that multiplies eponentiall b cell division. The number of bacteria grows continuousl at relative growth rate 1.386, that is, N = N 0 e 1.386t where N is the number of bacteria present after t hours and N 0 is the number of bacteria present at the start (t = 0). If we start with 2 bacteria, how man bacteria (to the nearest unit) will be present: (A) In 0.6 hour? (B) In 3. hours? SOLUTION Substituting N 0 = 2 into the preceding equation, we obtain N = 2e 1.386t The graph is shown in Figure 4.,000 N 0 Time (hours) t FIGURE 4 (A) Solve for N when t = 0.6: N = 2e 1.386(0.6) Use a calculator. = 7 bacteria (B) Solve for N when t = 3.: N = 2e 1.386(3.) Use a calculator. = 3,197 bacteria MATCHED PROBLEM 2 Refer to the eponential growth model for cholera in Eample 2. If we start with bacteria, how man bacteria (to the nearest unit) will be present (A) In 0.8 hour? (B) In 7.2 hours? EXAMPLE 3 Eponential Deca Cosmic-ra bombardment of the atmosphere produces neutrons, which in turn react with nitrogen to produce radioactive carbon-14 ( 14 C). Radioactive 14 C enters all living tissues through carbon dioide, which is first absorbed b plants.

54 14 C Section 2.4 Eponential Functions 99 As long as a plant or animal is alive, is maintained in the living organism at a constant level. Once the organism dies, however, 14 C decas according to the equation A = A 0 e t where A is the amount present after t ears and A 0 is the amount present at time t = 0. If 00 milligrams of 14 C is present in a sample from a skull at the time of death, how man milligrams will be present in the sample in (A) 1,000 ears? (B) 4,000 ears? Compute answers to two decimal places. SOLUTION Substituting A 0 = 00 in the deca equation, we have A = 00e t See the graph in Figure. A Milligrams Years 0,000 t FIGURE (A) Solve for A when t = 1,000: A = 00e (1,000) Use a calculator. = milligrams (B) Solve for A when t = 4,000: A = 00e (4,000) = 1.89 milligrams Use a calculator. MATCHED PROBLEM 3 Refer to the eponential deca model in Eample 3. How man milligrams of 14 C would have to be present at the beginning in order to have 2 milligrams present after 18,000 ears? Compute the answer to the nearest milligram. Eplore & Discuss 3 (A) On the same set of coordinate aes, graph the three deca equations A = A 0 e -0.3t, t Ú 0, for A 0 =, 20, and 30. (B) Identif an asmptotes for the three graphs in part (A). (C) Discuss the long-term behavior for the equations in part (A). If ou bu a new car, it is likel to depreciate in value b several thousand dollars during the first ear ou own it.you would epect the value of the car to decrease in each subsequent ear, but not b as much as in the previous ear. If ou drive the car long enough, its resale value will get close to zero. An eponential deca function will often be a good model of depreciation; a linear or quadratic function would not be suitable (wh?). We can use eponential regression on a graphing calculator to find the function of the form = ab that best fits a data set.

55 0 CHAPTER 2 Functions and Graphs EXAMPLE 4 Depreciation Table 2 gives the market value of a minivan (in dollars) ears after its purchase. Find an eponential regression model of the form = ab for this data set. Estimate the purchase price of the van. Estimate the value of the van ears after its purchase. Round answers to the nearest dollar. TABLE 2 Value ($) 1 12,7 2 9,4 3 8,11 4 6,84,22 6 4,48 SOLUTION Enter the data into a graphing calculator (Fig. 6A) and find the eponential regression equation (Fig. 6B). The estimated purchase price is 1 (0) = $14,9. The data set and the regression equation are graphed in Figure 6C. Using TRACE, we see that the estimated value after ears is $1, (A) (B) FIGURE 6 0 (C) MATCHED PROBLEM 4 Table 3 gives the market value of a luur sedan (in dollars) ears after its purchase. Find an eponential regression model of the form = ab for this data set. Estimate the purchase price of the sedan. Estimate the value of the sedan ears after its purchase. Round answers to the nearest dollar. TABLE 3 Value ($) 1 23, ,00 3 1, ,87 9,40 6 7,12 Compound Interest The fee paid to use another s mone is called interest. It is usuall computed as a percent (called interest rate) of the principal over a given period of time. If, at the end of a pament period, the interest due is reinvested at the same rate, then the interest earned as well as the principal will earn interest during the net pament period. Interest paid on interest reinvested is called compound interest, and ma be calculated using the following compound interest formula:

56 Section 2.4 Eponential Functions 1 If a principal P (present value) is invested at an annual rate r (epressed as a decimal) compounded m times a ear, then the amount A (future value) in the account at the end of t ears is given b A = Pa1 + r m b mt Compound interest formula Note: P could be replaced b A 0, but convention dictates otherwise. For given r and m, the amount A is equal to the principal P multiplied b the eponential function b t, where b = (1 + r>m) m. EXAMPLE SOLUTION Compound Growth If $1,000 is invested in an account paing % compounded monthl, how much will be in the account at the end of ears? Compute the answer to the nearest cent. We use the compound interest formula as follows: A = Pa1 + r m b mt The graph of = 1,000a b (12)() = $2, for 0 t 20 is shown in Figure 7. A = 1,000a b 12t Use a calculator. A $,000 $,000 0 Years 20 t FIGURE 7 MATCHED PROBLEM If ou deposit $,000 in an account paing 9% compounded dail, how much will ou have in the account in ears? Compute the answer to the nearest cent. Eplore & Discuss 4 Suppose that $1,000 is deposited in a savings account at an annual rate of %. Guess the amount in the account at the end of 1 ear if interest is compounded (1) quarterl, (2) monthl, (3) dail, (4) hourl. Use the compound interest formula to compute the amounts at the end of 1 ear to the nearest cent. Discuss the accurac of our initial guesses. The formula for compound interest is summarized on the net page for convenient reference.

57 2 CHAPTER 2 Functions and Graphs SUMMARY COMPOUND INTEREST FORMULA A = Pa1 + r m b mt where A = P = r = m = t = amount (future value) at the end of t ears principal (present value) annual rate, epressed as a decimal number of compounding periods per ear time in ears Answers to Matched Problems (A) 179 bacteria (B) 1,271,69 bacteria mg 4. Purchase price: $30,363; value after r: $2,864. $7, (A) $9,1.23 (B) $9,16.29 Eercise 2-4 A 1. Match each equation with the graph of f, g, h, or k in the figure. (A) = 2 (B) = (0.2) (C) = 4 (D) = ( 1 3 ) 2. Match each equation with the graph of f, g, h, or k in the figure. (A) = ( 1 4 ) (B) = (0.) (C) = (D) = 3 f g h k f g h k

58 Section 2.4 Eponential Functions 3 Graph each function in Problems 3 14 over the indicated interval. B 3. = ; [-2, 2] 4.. = ( 1 ) = - ; [-2, 2] f() = - ; [-2, 2] = -e - ; [-3, 3]. 11. = 0e 0.1 ; [-, ] g(t) = e -0.2t ; [-, ] 14. Simplif each epression in Problems (4 3 ) (2e 1.2t ) (3e -1.4 ) 2 In Problems 21 28, describe verball the transformations that can be used to obtain the graph of g from the graph of f (see Section 2-2) e - 3 e - 4 g() = -2 ; f() = 2 g() = 2-2 ; f() = 2 g() = ; f() = 3 g() = -3 ; f() = 3 g() = e + 1; f() = e g() = e - 2; f() = e g() = 2e -( + 2) ; f() = e - g() = 0.e -( - 1) ; f() = e Use the graph of f shown in the figure to sketch the graph of each of the following. (A) = f() - 1 (B) = f( + 2) (C) = 3f() - 2 (D) = 2 - f( - 3) 30. Use the graph of f shown in the figure to sketch the graph of each of the following. e e 1 - = 3 ; [-3, 3] = ( 1 3 ) = 3 - ; [-3, 3] g() = -3 - ; [-3, 3] = -e ; [-3, 3] = e 0.2 ; [-, ] f(t) = 0e -0.1t ; [-, ] (A) = f() + 2 (B) = f( - 3) (C) = 2f() - 4 (D) = 4 - f( + 2) Figure for 29 and 30 C In Problems 31 40, graph each function over the indicated interval. 31. f(t) = 2 t> ; [-30, 30] G(t) = 3 t>0 ; [-200, 200] = -3 + e 1 + ; [-4, 2] = 2 + e - 2 ; [-1, ] = e ƒƒ ; [-3, 3] = e -ƒƒ ; [-3, 3] C() = e + e - ; [-, ] 2 M() = e >2 + e ->2 ; [-, ] = e -2 ; [-3, 3] 40. = 2-2 ; [-3, 3] 41. Find all real numbers a such that a 2 = a -2. Eplain wh this does not violate the second eponential function propert in Theorem 2 on page Find real numbers a and b such that a Z b but a 4 = b 4. Eplain wh this does not violate the third eponential function propert in Theorem 2 on page 96. Solve each equation in Problems for = = = = = ( + 2) (1 - ) = (2-1) Solve each equation in Problems 49 2 for. (Remember: e Z 0 and e - Z 0. ) 49. ( - 3)e = e - = e e - = e - e = 0 Graph each function in Problems 3 6 over the indicated interval. 3. h() = (2 ); [-, 0] 4. m() = (3 - ); [0, 3]. N = 0 6. N = 1 + e -t; [0, ] e -t; [0, ] In Problems 7 60, approimate the real zeros of each function to two decimal places f() = f() = f() = f() =

59 4 CHAPTER 2 Functions and Graphs Applications In all problems involving das, a 36-da ear is assumed. 61. Finance. Suppose that $2,00 is invested at 7% compounded quarterl. How much mone will be in the account in 3 (A) 4 ear? (B) 1 ears? Compute answers to the nearest cent. 62. Finance. Suppose that $4,000 is invested at 6% compounded weekl. How much mone will be in the account in 1 (A) 2 ear? (B) ears? Compute answers to the nearest cent. 63. Finance. A person wishes to have $1,000 cash for a new car ears from now. How much should be placed in an account now, if the account pas 6.7% compounded weekl? Compute the answer to the nearest dollar. 64. Finance. A couple just had a bab. How much should the invest now at.% compounded dail in order to have $40,000 for the child s education 17 ears from now? Compute the answer to the nearest dollar. 6. Mone growth. BanQuote operates a network of Web sites providing real-time market data from leading financial providers. The following rates for 12-month certificates of deposit were taken from the Web sites: (A) Stonebridge Bank,.40% compounded monthl (B) DeepGreen Bank, 4.9% compounded dail (C) Provident Bank,.1% compounded quarterl Compute the value of $,000 invested in each account at the end of 1 ear. 66. Mone growth. Refer to Problem 6. The following rates for 60-month certificates of deposit were also taken from Ban- Quote Web sites: (A) Oriental Bank & Trust,.0% compounded quarterl (B) BMW Bank of North America,.12% compounded monthl (C) BankFirst Corporation, 4.86% compounded dail Compute the value of $,000 invested in each account at the end of ears. 67. Advertising. A compan is tring to introduce a new product to as man people as possible through television advertising in a large metropolitan area with 2 million possible viewers. A model for the number of people N (in millions) who are aware of the product after t das of advertising was found to be N = 2(1 - e t ) maimum number of boards an average emploee can be epected to produce in 1 da? 69. Sports salaries. Table 4 shows the average salaries for plaers in Major League Baseball (MLB) and the National Basketball Association (NBA) in selected ears since (A) Let represent the number of ears since 1990 and find an eponential regression model ( = ab ) for the average salar in MLB. Use the model to estimate the average salar (to the nearest thousand dollars) in 201. (B) The average salar in MLB in 2000 was million. How does this compare with the value given b the model of part (A)? How would the inclusion of the ear 2000 data affect the estimated average salar in 201? Eplain. TABLE 4 Average Salar (thousand $) Year MLB NBA ,062 1, ,1 2, ,724 2, ,300 4, ,633, Sports salaries. Refer to Table 4. (A) Let represent the number of ears since 1990 and find an eponential regression model ( = ab ) for the average salar in the NBA. Use the model to estimate the average salar (to the nearest thousand dollars) in 201. (B) The average salar in the NBA in 1997 was $2.2 million. How does this compare with the value given b the model of part (A)? How would the inclusion of the ear 1997 data affect the estimated average salar in 201? Eplain. 71. Marine biolog. Marine life is dependent upon the microscopic plant life that eists in the photic zone, a zone that goes to a depth where about 1% of the surface light remains. In some waters with a great deal of sediment, the photic zone ma go down onl 1 to 20 feet. In some murk harbors, the intensit of light d feet below the surface is given approimatel b I = I 0 e -0.23d Graph this function for 0 t 0. What value does N tend to as t increases without bound? 68. Learning curve. People assigned to assemble circuit boards for a computer manufacturing compan undergo on-the-job training. From past eperience it was found that the learning curve for the average emploee is given b N = 40(1 - e -0.12t ) where N is the number of boards assembled per da after t das of training. Graph this function for 0 t 30. What is the

60 Section 2.4 Logarithmic Functions What percentage of the surface light will reach a depth of (A) feet? (B) 20 feet? 72. Marine biolog. Refer to Problem 71. Light intensit I relative to depth d (in feet) for one of the clearest bodies of water in the world, the Sargasso Sea in the West Indies, can be approimated b I d I = I 0 e where is the intensit of light at the surface. What percentage of the surface light will reach a depth of (A) 0 feet? (B) 0 feet? 73. HIV/AIDS epidemic. The Joint United Nations Program on HIV/AIDS reported that HIV had infected 6 million people worldwide prior to Assume that number increases continuousl at a relative growth rate of 2%. (A) Write an equation that models the worldwide spread of HIV, letting 2006 be ear 0. (B) Based on the model, how man people (to the nearest million) had been infected prior to 2002? How man would be infected prior to 2014? (C) Sketch a graph of this growth equation from 2006 to HIV/AIDS epidemic. The Joint United Nations Program on HIV/AIDS reported that 2 million people worldwide had died of AIDS prior to Assume that number increases continuousl at a relative growth rate of 1%. (A) Write an equation that models total worldwide deaths from AIDS, letting 2006 be ear 0. (B) Based on the model, how man people (to the nearest million) had died from AIDS prior to 2002? How man would die from AIDS prior to 2014? (C) Sketch a graph of this growth equation from 2006 to World population growth. It took from the dawn of humanit to 1830 for the population to grow to the first billion people, just 0 more ears (b 1930) for the second billion, and 3 billion more were added in onl 60 more ears (b 1990). In 2006, the estimated world population was 6. billion with a relative growth rate of 1.14%. (A) Write an equation that models the world population growth, letting 2006 be ear 0. (B) Based on the model, what is the epected world population (to the nearest hundred million) in 201? In 2030? (C) Sketch a graph of the equation found in part (A). Cover the ears from 2006 through Population growth in Ethiopia. In 2006, the estimated population in Ethiopia was 7 million people with a relative growth rate of 2.3%. (A) Write an equation that models the population growth in Ethiopia, letting 2006 be ear 0. (B) Based on the model, what is the epected population in Ethiopia (to the nearest million) in 201? In 2030? (C) Sketch a graph of the equation found in part (A). Cover the ears from 2006 through Internet growth. The number of Internet hosts grew ver rapidl from 1994 to 2006 (Table ). (A) Let represent the number of ears since Find an eponential regression model ( = ab ) for this data set and estimate the number of hosts in 201 (to the nearest million). (B) Discuss the implications of this model if the number of Internet hosts continues to grow at this rate. TABLE Internet Hosts (Millions) Year TABLE 6 Year of Birth Life Epectanc Hosts Source: Internet Software Consortium 78. Life epectanc. Table 6 shows the life epectanc (in ears) at birth for residents of the United States from 1970 to 200. Let represent ears since Find an eponential regression model for this data and use it to estimate the life epectanc for a person born in 201. Section 2- LOGARITHMIC FUNCTIONS Inverse Functions Logarithmic Functions Properties of Logarithmic Functions Calculator Evaluation of Logarithms Applications Find the eponential function kes and e on our calculator. Close to these kes ou will find LOG and LN kes. The latter represent logarithmic functions, and each is closel related to the eponential function it is near. In fact, the eponential

61 6 CHAPTER 2 Functions and Graphs function and the corresponding logarithmic function are said to be inverses of each other. In this section we will develop the concept of inverse functions and use it to define a logarithmic function as the inverse of an eponential function. We will then investigate basic properties of logarithmic functions, use a calculator to evaluate them for particular values of, and appl them to real-world problems. Logarithmic functions are used in modeling and solving man tpes of problems. For eample, the decibel scale is a logarithmic scale used to measure sound intensit, and the Richter scale is a logarithmic scale used to measure the strength of the force of an earthquake. An important business application has to do with finding the time it takes mone to double if it is invested at a certain rate compounded a given number of times a ear or compounded continuousl. This requires the solution of an eponential equation, and logarithms pla a central role in the process. Inverse Functions Look at the graphs of f() = and g() = ƒ ƒ in Figure 1: 2 2 f() g() (A) f() 2 FIGURE 1 (B) g() 2 Because both f and g are functions, each domain value corresponds to eactl one range value. For which function does each range value correspond to eactl one domain value? This is the case onl for function f. Note that for the range value 2, the corresponding domain value is 4. For function g the range value 2 corresponds to both -4 and 4. Function f is said to be one-to-one. In general, DEFINITION One-to-One Functions A function f is said to be one-to-one if each range value corresponds to eactl one domain value. It can be shown that an continuous function that is either increasing or decreasing for all domain values is one-to-one. If a continuous function increases for some domain values and decreases for others, it cannot be one-to-one. Figure 1 shows an eample of each case. Eplore & Discuss 1 Graph f() = 2 and g() = 2. For a range value of 4, what are the corresponding domain values for each function? Which of the two functions is one-to-one? Eplain wh.

62 Section 2. Logarithmic Functions 7 Starting with a one-to-one function f, we can obtain a new function called the inverse of f as follows: DEFINITION Inverse of a Function If f is a one-to-one function, then the inverse of f is the function formed b interchanging the independent and dependent variables for f.thus, if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of the inverse of f. Note: If f is not one-to-one, then f does not have an inverse. A number of important functions in an librar of elementar functions are the inverses of other basic functions in the librar. In this course, we are interested in the inverses of eponential functions, called logarithmic functions. Logarithmic Functions If we start with the eponential function f defined b = 2 (1) and interchange the variables, we obtain the inverse of f: = 2 (2) We call the inverse the logarithmic function with base 2, and write = log 2 if and onl if = 2 We can graph = log b graphing = 2 2, since the are equivalent. An ordered pair of numbers on the graph of the eponential function will be on the graph of the logarithmic function if we interchange the order of the components. For eample, (3, 8) satisfies equation (1) and (8, 3) satisfies equation (2). The graphs of = 2 and = log 2 are shown in Figure 2. Note that if we fold the paper along the dashed line = in Figure 2, the two graphs match eactl. The line = is a line of smmetr for the two graphs. 2 2 or log 2 Eponential Function = 2 Logarithmic Function = FIGURE Ordered q C pairs S q reversed In general, since the graphs of all eponential functions of the form f() = b, b Z 1, b 7 0, are either increasing or decreasing (see Section 2-2), eponential functions have inverses.

63 8 CHAPTER 2 Functions and Graphs DEFINITION b Logarithmic Functions The inverse of an eponential function is called a logarithmic function. For b 7 0 and b Z 1, Logarithmic form Eponential form = log b is equivalent to = b log b The log to the base b of is the eponent to which b must be raised to obtain. [Remember: A logarithm is an eponent.] The domain of the logarithmic function is the set of all positive real numbers, which is also the range of the corresponding eponential function; and the range of the logarithmic function is the set of all real numbers, which is also the domain of the corresponding eponential function. Tpical graphs of an eponential function and its inverse, a logarithmic function, are shown in the figure in the margin. INSIGHT Because the domain of a logarithmic function consists of the positive real numbers, the entire graph of a logarithmic function lies to the right of the ais. In contrast, the graphs of polnomial and eponential functions intersect ever vertical line, and the graphs of rational functions intersect all but a finite number of vertical lines. The following eamples involve converting logarithmic forms to equivalent eponential forms, and vice versa. EXAMPLE 1 Logarithmic Eponential Conversions Change each logarithmic form to an equivalent eponential form: (A) log 2 = 2 (B) log 9 3 = 1 2 (C) log 2 ( 1 4 ) = -2 SOLUTION (A) log is equivalent to 2 = 2 2 = 2 1/2 (B) log 9 3 = 1 2 is equivalent to 3 = 9 (C) log 2 ( 1 4 ) = -2 is equivalent to 1 4 = 2-2 MATCHED PROBLEM 1 Change each logarithmic form to an equivalent eponential form: (A) log (B) log 4 2 = 1 (C) log 3 ( = ) = -2 EXAMPLE 2 Eponential Logarithmic Conversions Change each eponential form to an equivalent logarithmic form: (A) 64 = 4 3 (B) 6 = 236 (C) 1 8 = 2-3 SOLUTION (A) 64 = 4 3 is equivalent to log 4 64 = 3 (B) 6 = 236 is equivalent to log 36 6 = 1 2 (C) 1 8 = 2-3 is equivalent to log 2 ( 1 8 ) = -3

64 Section 2. Logarithmic Functions 9 MATCHED PROBLEM 2 Change each eponential form to an equivalent logarithmic form: (A) 49 = 7 2 (B) 3 = 29 (C) 1 3 = 3-1 To gain a little deeper understanding of logarithmic functions and their relationship to the eponential functions, we consider a few problems where we want to find, b, or in = log b, given the other two values. All values are chosen so that the problems can be solved eactl without a calculator. EXAMPLE 3 Solutions of the Equation = log b Find, b, or, as indicated. (A) Find : = log 4 16 (B) Find : log 2 = -3 (C) Find : = log 8 4 (D) Find b: log b 0 = 2 SOLUTION (A) = log is equivalent to 16 = Thus, = 2 (B) log is equivalent to = = -3. Thus, (C) = log 8 4 is equivalent to = = 1 8 Thus, 4 = 8 or 2 2 = = 2 = 2 3 (D) log is equivalent to 0 = b 2 b 0 = 2. Thus, b = Recall that b cannot be negative. MATCHED PROBLEM 3 Find, b, or, as indicated. (A) Find : = log 9 27 (C) Find b: log b 1,000 = 3 (B) Find : log 3 = -1 Properties of Logarithmic Functions Logarithmic functions have man powerful and useful properties. We list eight basic properties in Theorem 1. THEOREM 1 PROPERTIES OF LOGARITHMIC FUNCTIONS If b, M, and N are positive real numbers, b Z 1, and p and are real numbers, then 1. log b 1 = 0. log b MN = log b M + log b N 2. log b b = 1 M 6. log b N = log b M - log b N 3. log b b = 7. log b M p = p log b M 4. b log b =, log b M = log b N if and onl if M = N The first four properties in Theorem 1 follow directl from the definition of a logarithmic function. Here we will sketch a proof of propert. The other properties are established in a similar wa. Let u = log b M and v = log b N

65 1 CHAPTER 2 Functions and Graphs Or, in equivalent eponential form, M = b u and N = b v Now, see if ou can provide reasons for each of the following steps: log b MN = log b b u b v = log b b u + v = u + v = log b M + log b N EXAMPLE 4 Using Logarithmic Properties (A) w log b z = log b w - log b z = log b w + log b - (log b + log b z) = log b w + log b - log b - log b z (B) log b(w) 3/ = 3 log b w = (C) e log e b = e log e b = b (D) log e log e b = log e(b logb ) log e b 3 (log b w + log b ) = (log b )(log e b) log e b = log b MATCHED PROBLEM 4 Write in simpler forms, as in Eample 4. (A) (B) (C) (D) log 2 2 u log 2 b log b a R 2/3 R log b ST S b log 2 b The following eamples and problems, although somewhat artificial, will give ou additional practice in using basic logarithmic properties. EXAMPLE Solving Logarithmic Equations Find so that 3 2 log b log b 8 + log b 2 = log b SOLUTION 3 2 log b log b 8 + log b 2 = log b log b 4 3/2 - log b 8 2/3 + log b 2 = log b log b 8 - log b 4 + log b 2 = log b 8 # 2 log b 4 = log b log b 4 = log b = 4 Propert 7 Properties and 6 Propert 8 MATCHED PROBLEM Find so that 3 log b log b 2 - log b 20 = log b.

66 Section 2. Logarithmic Functions 111 EXAMPLE 6 Solving Logarithmic Equations Solve: log + log ( + 1) = log 6. SOLUTION log + log ( + 1) = log 6 log [( + 1)] = log 6 Propert ( + 1) = 6 Propert = 0 Solve b factoring. ( + 3)( - 2) = 0 = -3, 2 We must eclude = -3, since the domain of the function log ( + 1) is 7-1 or (-1, q ); hence, = 2 is the onl solution. MATCHED PROBLEM 6 Solve: log 3 + log 3 ( - 3) = log 3. Eplore & Discuss 2 Discuss the relationship between each of the following pairs of epressions. If the two epressions are equivalent, eplain wh. If the are not, give an eample. (A) log b M - log b N; log b M log b N M (B) log b M - log b N; log b N (C) log b M + log b N; log b MN (D) log b M + log b N; log b (M + N) Calculator Evaluation of Logarithms Of all possible logarithmic bases, the base e and the base are used almost eclusivel. Before we can use logarithms in certain practical problems, we need to be able to approimate the logarithm of an positive number either to base or to base e. And conversel, if we are given the logarithm of a number to base or base e,we need to be able to approimate the number. Historicall, tables were used for this purpose, but now calculators make computations faster and far more accurate. Common logarithms (also called Briggsian logarithms) are logarithms with base. Natural logarithms (also called Napierian logarithms) are logarithms with base e. Most calculators have a ke labeled log (or LOG ) and a ke labeled ln (or LN ). The former represents a common (base ) logarithm and the latter a natural (base e) logarithm. In fact, log and ln are both used etensivel in mathematical literature, and whenever ou see either used in this book without a base indicated, the will be interpreted as follows: Common logarithm: Natural logarithm: log means log ln means log e Finding the common or natural logarithm using a calculator is ver eas. On some calculators, ou simpl enter a number from the domain of the function and press LOG or LN. On other calculators, ou press either LOG or LN, enter a number from the domain, and then press ENTER. Check the user s manual for our calculator. EXAMPLE 7 Calculator Evaluation of Logarithms Use a calculator to evaluate each to si decimal places: (A) log 3,184 (B) ln (C) log(-3.24)

67 112 CHAPTER 2 Functions and Graphs SOLUTION (A) (B) log 3,184 = ln = (C) log(-3.24) = Error* 3.24 is not in the domain of the log function. MATCHED PROBLEM 7 Use a calculator to evaluate each to si decimal places: (A) log (B) ln (C) ln( ) We now turn to the second problem mentioned previousl: Given the logarithm of a number, find the number.we make direct use of the logarithmic eponential relationships, which follow from the definition of logarithmic function given at the beginning of this section. log is equivalent to ln is equivalent to e EXAMPLE 8 Solving log b for logarithm: (A) log = (B) ln = Find to four decimal places, given the indicated SOLUTION (A) log = Change to equivalent eponential form. (B) = = ln = = e =.8699 Evaluate with a calculator. Change to equivalent eponential form. Evaluate with a calculator. MATCHED PROBLEM 8 Find to four decimal places, given the indicated logarithm: (A) ln = (B) log = EXAMPLE 9 Solving Eponential Equations Solve for to four decimal places: (A) = 2 (B) e = 3 (C) 3 = 4 * Some calculators use a more advanced definition of logarithms involving comple numbers and will displa an ordered pair of real numbers as the value of log(-3.24). You should interpret such a result as an indication that the number entered is not in the domain of the logarithm function as we have defined it.

68 SOLUTION (A) (B) (C) = 2 log = log 2 = log 2 e = 3 ln e = ln 3 = ln 3 3 = 4 = 0.30 = log 3 = log 4 log 3 = log 4 = log 4 log 3 = Section 2. Logarithmic Functions 113 Take common logarithms of both sides. Propert 3 Use a calculator. To four decimal places Take natural logarithms of both sides. Propert 3 Use a calculator. To four decimal places Take either natural or common logarithms of both sides. (We choose common logarithms.) Propert 7 Solve for. Use a calculator. To four decimal places Eponential equations can also be solved graphicall b graphing both sides of an equation and finding the points of intersection. Figure 3 illustrates this approach for the equations in Eample (A) (B) 1 e (C) FIGURE 3 Graphical solution of eponential equations MATCHED PROBLEM 9 Solve for to four decimal places: (A) = 7 (B) e = 6 (C) 4 = Eplore & Discuss 3 Discuss how ou could find = log 38.2 using either natural or common logarithms on a calculator. [Hint: Start b rewriting the equation in eponential form.] REMARK In the usual notation for natural logarithms, the simplifications of Eample 4, parts (C) and (D) on page 114, become e ln b = b ln and ln b = log b With these formulas we can change an eponential function with base b, or a logarithmic function with base b, to epressions involving eponential or logarithmic functions, respectivel, to the base e. Such change-of-base formulas are useful in calculus.

69 114 CHAPTER 2 Functions and Graphs APPLICATIONS A convenient and easil understood wa of comparing different investments is to use their doubling times the length of time it takes the value of an investment to double. Logarithm properties, as ou will see in Eample, provide us with just the right tool for solving some doubling-time problems. EXAMPLE SOLUTION Doubling Time for an Investment How long (to the net whole ear) will it take mone to double if it is invested at 20% compounded annuall? We use the compound interest formula discussed in Section 2-4: A = Pa1 + r mt Compound interest m b A = 2p; The problem is to find t, given r = 0.20, m = 1, and that is, 2P = P( ) t 2 = 1.2 t 1.2 t = 2 ln 1.2 t = ln 2 t ln 1.2 = ln 2 t = ln 2 ln 1.2 Solve for t b taking the natural or common logarithm of both sides (we choose the natural logarithm). Propert 7 Use a calculator. L 3.8 ears [Note: (ln 2)/(ln 1.2) Z ln 2 ln 1.2] L 4 ears To the net whole ear When interest is paid at the end of 3 ears, the mone will not be doubled; when paid at the end of 4 ears, the mone will be slightl more than doubled. Eample can also be solved graphicall b graphing both sides of the equation 2 = 1.2 t, and finding the intersection point (Fig. 4) FIGURE 4 1 = 1.2, 2 = 2 MATCHED PROBLEM How long (to the net whole ear) will it take mone to triple if it is invested at 13% compounded annuall? It is interesting and instructive to graph the doubling times for various rates compounded annuall. We proceed as follows: A = P(1 + r) t 2P = P(1 + r) t (1 + r) t = 2 2 = (1 + r) t ln(1 + r) t = ln 2 t ln(1 + r) = ln 2 t = ln 2 ln(1 + r)

70 Section 2. Logarithmic Functions 11 Figure shows the graph of this equation (doubling time in ears) for interest rates compounded annuall from 1 to 70% (epressed as decimals). Note the dramatic change in doubling time as rates change from 1 to 20% (from 0.01 to 0.20). t Years t ln 2 ln (1 r) Rate compounded annuall FIGURE r Among increasing function, the logarithmic functions (with bases b 7 1) increase much more slowl for large values of than either eponential or polnomial functions. When a visual inspection of the plot of a data set indicates a slowl increasing function, a logarithmic function often provides a good model. We use logarithmic regression on a graphing calculator to find the function of the form = a + b ln that best fits the data. EXAMPLE 11 Home Ownership Rates The U.S. Census Bureau published the data in Table 1 on home ownership rates. Let represent time in ears with = 0 representing Use logarithmic regression to find the best model of the form = a + b ln for the home ownership rate as a function of time. Use the model to predict the home ownership rate in the United States in 201 (to the nearest tenth of a percent). TABLE 1 Home Ownership Rates Year Rate (%) SOLUTION Enter the data in a graphing calculator (Fig. 6A) and find the logarithmic regression equation (Fig. 6B). The data set and the regression equation are graphed in Figure 6C. Using TRACE, we predict that the home ownership rate in 201 would be 69.4% (A) (B) FIGURE 6 40 (C)

71 116 CHAPTER 2 Functions and Graphs MATCHED PROBLEM 11 The home ownership rate in 200 was 69.1%. Add this point to the data of Table 1, and, with = 0 representing 1900, use logarithmic regression to find the best model of the form = a + b ln for the home ownership rate as a function of time. Use the model to predict the home ownership rate in the United States in 201 (to the nearest tenth of a percent). CAUTION: Note that in Eample 11 we let = 0 represent If we let = 0 represent 1940, for eample, we would obtain a different logarithmic regression equation, but the prediction for 201 would be the same. We would not let = 0 represent 190 (the first ear in Table 1) or an later ear, because logarithmic functions are undefined at 0. Answers to Matched Problems 1 9 = (A) 9 = 3 2 (B) 2 = 4 1/2 (C) 2.(A) log 7 49 = 2 (B) log 9 3 = 1 2 (C) log 3 ( 1 3 ) = -1 3.(A) = 3 2 (B) = 1 3 (C) b = 4.(A) log b R - log b S - log b T (B) 3 (log b R - log b S) (C) b u (D) log b. = 2 6. = 7.(A) (B) (C) Not defined 8.(A) (B) (A) (B) (C) r % Eercise 2- A For Problems 1 6, rewrite in equivalent eponential form. 1. log 3 27 = 3 2. log 2 32 = 3. log 1 = 0 4. log e 1 = 0. log 4 8 = log 9 27 = 3 2 For Problems 7 12, rewrite in equivalent logarithmic form = = = 4 3/2. 2/3 9 = A = b u 12. M = b For Problems 13 24, evaluate without a calculator. 13. log log e 1 1. log e e 16. log 17. log log log log 21. log log log 1, log 6 36 B For Problems 2 30, write in terms of simpler forms, as in Eample P log b Q 26. log b FG 27. log b L 28. log b w 1 3 p log 3 q log 3 P log 3 R For Problems 31 42, find,, or b without a calculator. 31. log 3 = log 2 = log 7 49 = 34. log 3 27 = 3. log b -4 = log b e -2 = log 4 = log 2 = log 1/3 9 = 40. log 49 ( 1 7 ) = 41. log b 1,000 = log b 4 = 2 3

72 Section 2. Logarithmic Functions 117 In Problems 43 2, discuss the validit of each statement. If the statement is alwas true, eplain wh. If not, give a countereample. 43. Ever polnomial function is one-to-one. 44. Ever polnomial function of odd degree is one-to-one. 4. If g is the inverse of a function f, then g is one-to-one. 46. The graph of a one-to-one function intersects each vertical line eactl once. 47. The inverse of f() = 2 is g() = / The inverse of f() = 2 is g() = If b 7 0 and b Z 1, then the graph of = log b is increasing. 0. If b 7 1, then the graph of = log b is increasing. 1. If f is one-to-one, then the domain of f is equal to the range of f. 2. If g is the inverse of a function f, then f is the inverse of g. Find in Problems log b = 2 3 log b log b 9 - log b 6 4. log b = 2 3 log b log b 2 - log b 3. log b = 3 2 log b log b log b 2 6. log b = 3 log b log b 2 - log b log b + log b ( - 4) = log b log b ( + 2) + log b = log b log ( - 1) - log ( + 1) = log ( + 6) - log ( - 3) = 1 Graph Problems 61 and 62 b converting to eponential form first. 61. = log 2 ( - 2) 62. = log 3 ( + 2) 63. Eplain how the graph of the equation in Problem 61 can be obtained from the graph of = log 2 using a simple transformation (see Section 2-2). 64. Eplain how the graph of the equation in Problem 62 can be obtained from the graph of = log 3 using a simple transformation (see Section 2-2). 6. What are the domain and range of the function defined b = 1 + ln( + 1)? 66. What are the domain and range of the function defined b = log( - 1) - 1? For Problems 67 and 68, evaluate to five decimal places using a calculator. Applications C 67. (A) log 3,27.2 (B) log (C) ln (D) ln (A) log (B) log (C) ln 40,27 (D) ln For Problems 69 and 70, find to four decimal places. 69. (A) log = (B) log = (C) ln = (D) ln = (A) log = (B) log = (C) ln = (D) ln = For Problems 71 78, solve each equation to four decimal places. 71. = = e = e = = = t = t = 2 ƒ ƒ ƒ ƒ Graph Problems using a calculator and point-b-point plotting. Indicate increasing and decreasing intervals. 79. = ln 80. = -ln 81. = ln 82. = ln 83. = 2 ln( + 2) 84. = 2 ln = 4 ln = 4 ln( - 3) 87. Eplain wh the logarithm of 1 for an permissible base is Eplain wh 1 is not a suitable logarithmic base. 89. Write log - log c = 0.8 in an eponential form that is free of logarithms. 90. Write log e - log e 2 = 0.2t in an eponential form that is free of logarithms. 91. Let p() = ln, q() = 1, and r() =. Use a graphing calculator to draw graphs of all three functions in the same viewing window for Discuss what it means for one function to be larger than another on an interval, and then order the three functions from largest to smallest for Let p() = log, q() = 1, 3 and r() =. Use a graphing calculator to draw graphs of all three functions in the same viewing window for Discuss what it means for one function to be smaller than another on an interval, and then order the three functions from smallest to largest for Doubling time. In its first ears the Gabelli Growth Fund produced an average annual return of 21.36%. Assume that mone invested in this fund continues to earn 21.36% compounded annuall. How long will it take mone invested in this fund to double? 94. Doubling time. In its first ears the Janus Fleible Income Fund produced an average annual return of 9.8%. Assume that mone invested in this fund continues to earn 9.8% compounded annuall. How long will it take mone invested in this fund to double?

73 118 CHAPTER 2 Functions and Graphs 9. Investing. How man ears (to two decimal places) will it take $1,000 to grow to $1,800 if it is invested at 6% compounded quarterl? Compounded dail? 96. Investing. How man ears (to two decimal places) will it take $,000 to grow to $7,00 if it is invested at 8% compounded semiannuall? Compounded monthl? 97. Suppl and demand. A cordless screwdriver is sold through a national chain of discount stores. A marketing compan established price demand and price suppl tables (Tables 1 and 2), where is the number of screwdrivers people are willing to bu and the store is willing to sell each month at a price of p dollars per screwdriver. (A) Find a logarithmic regression model ( = a + b ln ) for the data in Table 1. Estimate the demand (to the nearest unit) at a price level of $0. TABLE 1 Price Demand p D() ($) 1, , , ,000 6,000 3 (B) Find a logarithmic regression model ( = a + b ln ) for the data in Table 2. Estimate the suppl (to the nearest unit) at a price level of $0. TABLE 2 Price Suppl p S() ($) 1, , , ,000 38, (C) Does a price level of $0 represent a stable condition, or is the price likel to increase or decrease? Eplain. 98. Equilibrium point. Use the models constructed in Problem 97 to find the equilibrium point. Write the equilibrium price to the nearest cent and the equilibrium quantit to the nearest unit. 99. Sound intensit: decibels. Because of the etraordinar range of sensitivit of the human ear (a range of over 1,000 million millions to 1), it is helpful to use a logarithmic scale, rather than an absolute scale, to measure sound intensit over this range. The unit of measure is called the decibel, after the inventor of the telephone,aleander Graham Bell. If we let N be the number of decibels, I the power of the sound in question (in watts per square centimeter), and I 0 the power of sound just below the threshold of hearing (approimatel -16 watt per square centimeter), then I = I 0 N/ Show that this formula can be written in the form N = log I I 0 0. Sound intensit: decibels. Use the formula in Problem 99 (with I 0 = -16 W/cm 2 ) to find the decibel ratings of the following sounds: (A) Whisper: -13 W/cm 2 (B) Normal conversation: 3.16 * - W/cm 2 (C) Heav traffic: -8 W/cm 2 (D) Jet plane with afterburner: -1 W/cm 2 1. Agriculture. Table 3 shows the ield (in bushels per acre) and the total production (in millions of bushels) for corn in the United States for selected ears since 190. Let represent ears since Find a logarithmic regression model ( = a + b ln ) for the ield. Estimate (to one decimal place) the ield in 201. TABLE 3 United States Corn Production Yield (bushels Total Production Year per acre) (million bushels) , , , , , , Agriculture. Refer to Table 3. Find a logarithmic regression model ( = a + b ln ) for the total production. Estimate (to the nearest million) the production in World population. If the world population is now 6. billion people and if it continues to grow at an annual rate of 1.14% compounded continuousl, how long (to the nearest ear) will it take before there is onl 1 square ard of land per person? (The Earth contains approimatel 1.68 * 14 square ards of land.) 4. Archaeolog: carbon-14 dating. The radioactive carbon-14 ( 14 C) in an organism at the time of its death decas according to the equation A = A 0 e t 14 C where t is time in ears and A 0 is the amount of present at time t = 0. (See Eample 3 in Section 2-4.) Estimate the age of a skull uncovered in an archaeological site if % of the original amount of 14 C is still present. [Hint: Find t such that A = 0.1A 0.

74 Chapter 2 Review 119 CHAPTER 2 REVIEW Important, Terms, Smbols, and Concepts 2-1 Functions Point-b-point plotting ma be used to sketch the graph of an equation in two variables: Plot enough points from its solution set in a rectangular cordinate sstem so that the total graph is apparent and then connect these points with a smooth curve. A function is a correspondence between two sets of elements such that to each element in the first set there corresponds one and onl one element in the second set. The first set is called the domain and the set of corresponding elements in the second set is called the range. If is a placeholder for the elements in the domain of a function, then is called the independent variable or the input. If is a placeholder for the elements in the range, then is called the dependent variable or the output. If in an equation in two variables we get eactl one ouput for each input, then the equation specifies a function. The graph of such a function is just the graph of the specifing equation. If we get more than one output for a given input, then the equation does not specif a function. The vertical-line test can be used to determine whether or not an equation in two variables specifies a function (Theorem 1, p. 1). The functions specified b equations of the form = m + b, where m Z 0, are called linear functions. Functions specified b equations of the form = b are called constant functions. If a function is specified b an equation and the domain is not indicated, we agree to assume that the domain is the set of all inputs that produce outputs that are real numbers. The smbol f() represents the element in the range of f that corresponds to the element of the domain. Break-even and profit loss analsis use a cost function C and a revenue function R to determine when a compan will have a loss (R 6 C), will break even (R = C), or will have a profit (R 7 C). Tpical cost, revenue, profit, and price demand functions are given on page. 2-2 Elementar Functions: Graphs and Transformations The graphs of si basic elementar functions (the identit function, the square and cube functions, the square root and cube root functions, and the absolute value function) are shown on page 6. Performing an operation on a function produces a transformation of the graph of the function. The basic graph transformations, vertical and horizontal translations (shifts), reflection in the ais, and vertical stretches and shrinks, are summarized on page 69. A piecewise-defined function is a function whose definition involves more than one formula. 2-3 Quadratic Functions Eamples E. 1, p. 47 E. 2, p. 0 E. 3, p. 2 E., p. 4 E. 4, p. 3 E. 6, p. E. 7, p.6 E. 1, p. 64 E. 2, p. 66 E. 3, p. 67 E. 4, p. 68 E., p. 69 E. 6, p. 70 If a, b, and c are real numbers with a Z 0, then the function f() = a 2 + b + c Standard form is a quadratic function in standard form and its graph is a parabola. The quadratic formula E. 1, p. 77 = - b ; 2b2-4ac 2a b 2-4ac Ú 0 can be used to find the intercepts of a quadratic function. Completing the square in the standard form of a quadratic function produces the verte form f() = a( - h) 2 + k Verte form From the verte form of a quadratic function, we can read off the verte, ais of smmetr, maimum or minimum, and range, and can easil sketch the graph (page 81). If a revenue function R() and a cost function C() intersect at a point ( 0, 0 ), then both this point and its coordinate 0 are referred to as break-even points. E. 2, p. 81 E. 3, p. 83 E. 4, p. 84

75 120 CHAPTER 2 Functions and Graphs Quadratic regression on a graphing calculator produces the function of the form = a 2 + b + c E., p. 86 that best fits a data set. A quadratic function is a special case of a polnomial function, that is, a function that can be written in the form for n a nonnegative integer called the degree of the polnomial. The coefficients a 0, a 1,..., a n are real numbers with a n Z 0. The domain of a polnomial function is the set of all real numbers. Graphs of representative polnomial functions are shown on page 87 and inside the front cover. The graph of a polnomial function of degree n can intersect the ais at most n times. The graph of a polnomial function has no sharp corners and is continuous, that is, it has no holes or breaks. A rational function is an function that can be written in the form where n() and d() are polnomials. The domain is the set of all real numbers such that d() Z 0. Graphs of representative rational functions are shown on page 88 and inside the front cover. Unlike polnomial functions, a rational function can have vertical asmptotes [but not more than the degree of the denominator d() ] and at most one horizontal asmptote. 2-4 Eponential Functions f() = a n n + a n - 1 n a 1 + a 0 f() = n() d() d() Z 0 An eponential function is a function of the form f() = b where b Z 1 is a positive constant called the base. The domain of f is the set of all real numbers and the range is the set of positive real numbers. The graph of an eponential function is continuous, passes through (0, 1), and has the ais as a horizontal asmptote. If b 7 1, then b increases as increases; if 0 6 b 6 1, then b decreases as increases (Theorem 1, p. 9). Eponential functions obe the familiar laws of eponents and satisf additional properties (Theorem 2, p. 96). The base that is used most frequentl in mathematics is the irrational number e L Eponential functions can be used to model population growth and radioactive deca. Eponential regression on a graphing calculator produces the function of the form = ab that best fits a data set. Eponential functions are used in computations of compound interest: A = Pa1 + r m b mt Compound interest formula E. 1, p. 9 E. 2, p. 98 E. 3, p. 98 E. 4, p. 0 E., p. 1 (see summar on page 2). 2- Logarithmic Functions A function is said to be one-to-one if each range value corresponds to eactl one domain value. The inverse of a one-to-one function f is the function formed b interchanging the independent and dependent variables of f. That is, (a, b) is a point on the graph of f if and onl if (b, a) is a point on the graph of the inverse of f. A function that is not one-to-one does not have an inverse. The inverse of the eponential function with base b is called the logarithmic function with base b, denoted = log b. The domain of log b is the set of all positive real numbers (which is the range of b ), and the range of log is the set of all real numbers (which is the domain of b b ). Because log is the inverse of the function b b, Logarithmic form = log b Eponential form is equivalent to = b E. 1, p. 8 E. 2, p. 8 E. 3, p. 9

76 Chapter 2 Review 121 Properties of logarithmic functions can be obtained from corresponding properties of epontial functions (Theorem 1, p. 9). Logarithms to the base are called common logarithms, often denoted simpl b log. Logarithms to the base e are called natural logarithms, often denoted b ln. Logarithms can be used to find an investment s doubling time the length of time it takes for the value of an investment to double. Logarithmic regression on a graphing calculator produces the function of the form = a + b ln that best fits a data set. E. 4, p.1 E., p. 1 E. 6, p. 111 E. 7, p. 111 E. 8, p. 112 E. 9, p. 112 E., p. 114 E. 11, p. 11 REVIEW EXERCISE Work through all the problems in this chapter review and check our answers in the back of the book. Answers to all review problems are there along with section numbers in italics to indicate where each tpe of problem is discussed. Where weaknesses show up, review appropriate sections in the tet. A In Problems 1 3, use point-b-point plotting to sketch the graph of each equation. 1. = = = Indicate whether each graph specifies a function: (A) (B) 6. Write in logarithmic form using base e: u = e v. 7. Write in logarithmic form using base : =. 8. Write in eponential form using base e: ln M = N. 9. Write in eponential form using base : log u = v. Simplif Problems and a eu e -u b u Solve Problems for eactl without using a calculator. 12. log 3 = log 36 = log 2 16 = (C) (D) Solve Problems 1 18 for to three decimal places. 1. = e = 03, log = ln = Use the graph of function f in the figure to determine (to the nearest integer) or as indicated. (A) = f(0) (B) 4 = f() (C) = f(3) (D) 3 = f() (E) = f(-6) (F) -1 = f() f() 7 7. For f() = 2-1 and g() = 2-2, find: (A) f(-2) + g(-1) (B) f(0) # g(4) g(2) (C) (D) f(3) f(3) g(2) Figure for 19

77 122 CHAPTER 2 Functions and Graphs 20. Sketch a graph of each of the functions in parts (A) (D) using the graph of function f in the figure below. (A) = -f() (B) = f() + 4 (C) = f( - 2) (D) = -f( + 3) - 3 f() Figure for Complete the square and find the standard form for the quadratic function Then write a brief verbal description of the relationship between the graph of f and the graph of = Match each equation with a graph of one of the functions f, g, m, or n in the figure. f m (A) = ( - 2) 2-4 (C) = -( - 2) f() = Figure for 22 g n 2. The three points in the table are on the graph of the indicated function f. Do these three points provide sufficient in- (B) = -( + 2) (D) = ( + 2) Referring to the graph of function f in the figure for Problem 11 and using known properties of quadratic functions, find each of the following to the nearest integer: (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range (E) Increasing interval (F) Decreasing interval 24. Consider the set S of people living in the town of Newville. Which of the following correspondences specif a function? Eplain. (A) Each person in Newville (input) is paired with his or her mother (output). (B) Each person in Newville (input) is paired with his or her children (output). formation for ou to sketch the graph of = f()? Add more points to the table until ou are satisfied that our sketch is a good representation of the graph of = f() on the interval [-, ] f() In Problems 26 29, each equation specifies a function. Determine whether the function is linear, quadratic, constant, or none of these. 26. = = = = 8 + 2( - 4) 2 Solve Problems for eactl without using a calculator. 30. log( + ) = log(2-3) ln( - 1) = ln( 2 - ) e 2 = e = e = 3e 3. log 1/3 9 = 36. log 37. log 9 = 3 8 = -3 2 Solve Problems for to four decimal places. 38. = 3(e = 230( ) 40. log = ln = = 7(3 ) = e ,000 = 4,000(1.08 ) = = log = log Find the domain of each function: 2-3 (A) f() = (B) g() = The function g is defined b g() = Translate into a verbal definition. 0. Find the verte form for f() = and then find the intercepts, the verte, the maimum or minimum, and the range. 1. Let f() = e - 1 and g() = ln ( + 2). Find all points of intersection for the graphs of f and g. Round answers to two decimal places. In Problems 2 and 3, use point-b-point plotting to sketch the graph of each function f() = f() = If f() = + 1, find and simplif each of the following in Problems f(f(0)). f(f(-1)) 6. f(2-1) 7. f(4 - ) 8. Let f() = 3-2. Find (A) f(2) (B) f(2 + h) (C) f(2 + h) - f(2) f() = 6-3 (D) f(2 + h) - f(2) h

78 Chapter 2 Review Let f() = Find (A) f(a) (B) f(a + h) (C) f(a + h) - f(a) (D) f(a + h) - f(a) h 60. Eplain how the graph of m() = -ƒ - 4 is related to the graph of =. the graph of. 61. Eplain how the graph of g() = is related to 62. The following graph is the result of appling a sequence of transformations to the graph of = 2. Describe the transformations verball and write an equation for the graph. Figure for The graph of a function f is formed b verticall stretching the graph of = 1 b a factor of 2, and shifting it to the left 3 units and down 1 unit. Find an equation for function f and graph it for - and -. In Problems 64 71, discuss the validit of each statement. If the statement is alwas true, eplain wh. If not, give a countereample. 64. Ever polnomial function is a rational function. 6. Ever rational function is a polnomial function. 66. The graph of ever rational function has at least one vertical asmptote. 67. The graph of ever eponential function has a horizontal asmptote. 68. The graph of ever logarithmic function has a vertical asmptote. 69. There eists a logarithmic function that has both a vertical and horizontal asmptote. 70. There eists a rational function that has both a vertical and horizontal asmptote. 71. There eists an eponential function that has both a vertical and horizontal asmptote. Graph Problems over the indicated interval. Indicate increasing and decreasing intervals. 72. = 2-1 ; [-2, 4] 73. f(t) = e -0.08t ; t Ú = ln( + 1); (-1, ] 7. Sketch the graph of f for Ú if 0 20 f() = e if Sketch the graph of g for Ú if 0 g() = c if if Write an equation for the graph shown in the form = a( - h) 2 + k, where a is either -1 or +1 and h and k are integers. Figure for Given f() = , find the following algebraicall (to three decimal places) without referring to a graph: (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range 79. Graph f() = in a graphing calculator and find the following (to three decimal places) using TRACE and appropriate built-in commands: (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range C 80. Noting that p = and 12 = eplain wh the calculator results shown here are obvious. Discuss similar connections between the natural logarithmic function and the eponential function with base e. Solve Problems eactl without using a calculator. 81. log - log 3 = log 4 - log( + 4) 82. ln(2-2) - ln( - 1) = ln 83. ln( + 3) - ln = 2 ln log 3 2 = 2 + log 9 8. Write ln = -t + ln c in an eponential form free of logarithms. Then solve for in terms of the remaining variables. 86. Eplain wh 1 cannot be used as a logarithmic base. 87. The following graph is the result of appling a sequence of transformations to the graph of = 1 3. Describe the transformations verball, and write an equation for the graph. Figure for 87

79 124 CHAPTER 2 Functions and Graphs 88. Given G() = , find the following algebraicall (to three decimal places) without the use of a graph: (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range (E) Increasing and decreasing intervals 89. Graph G() = in a standard viewing window. Then find each of the following (to three decimal places) using appropriate commands. (A) Intercepts (B) Verte (C) Maimum or minimum (D) Range (E) Increasing and decreasing intervals Applications In all problems involving das, a 36-da ear is assumed. 90. Egg consumption. The table shows the annual per capita consumption of eggs in the United States. Year A mathematical model for this data is given b f() = where = 0 corresponds to (A) Complete the following table. Number of Eggs Consumption f() (B) Graph = f() and the data in the same coordinate sstem. (C) Use the modeling function f to estimate the per capita egg consumption in (D) Based on the information in the table, write a brief description of egg consumption from 1980 to Egg consumption. Using quadratic regression on a graphing calculator, show that the quadratic function that best fits the data on egg consumption in Problem 90 is f() = (coefficients rounded to three significant digits). 92. Electricit rates. The table shows the electricit rates charged b Easton Utilities in the summer months. (A) Write a piecewise definition of the monthl charge S() (in dollars) for a customer who uses kwh in a summer month. (B) Graph S(). Energ Charge (June September) $3.00 for the first 20 kwh or less.70 per kwh for the net 180 kwh 3.46 per kwh for the net 800 kwh 2.17 per kwh for all over 1,000 kwh 93. Mone growth. Provident Bank of Cincinnati, Ohio recentl offered a certificate of deposit that paid.3% compounded quarterl. If a $,000 CD earns this rate for ears, how much will it be worth? 94. Mone growth. Capital One Bank of Glen Allen, Virginia recentl offered a certificate of deposit that paid 4.82% compounded dail. If a $,000 CD earns this rate for ears, how much will it be worth? 9. Mone growth. How long will it take for mone invested at 6.9% compounded monthl to triple? 96. Mone growth. How long will it take for mone invested at 7.39% compounded dail to double? 97. Break-even analsis. The research department in a compan that manufactures AM/FM clock radios established the following price demand, cost, and revenue functions: p() = C() = R() = p() = (0-1.2) Price demand function Cost function Revenue function where is in thousands of units, and C() and R() are in thousands of dollars. All three functions have domain (A) Graph the cost function and the revenue function simutaneousl in the same coordinate sstem. (B) Determine algebraicall when R = C. Then, with the aid of part (A), determine when R 6 C and R 7 C to the nearest unit. (C) Determine algebraicall the maimum revenue (to the nearest thousand dollars) and the output (to the nearest unit) that produces the maimum revenue. What is the wholesale price of the radio (to the nearest dollar) at this output? 98. Profit loss analsis. Use the cost and revenue functions from Problem 97. (A) Write a profit function and graph it in a graphing calculator.

80 Chapter 2 Review 12 (B) Determine graphicall when P = 0, P 6 0, and P 7 0 to the nearest unit. (C) Determine graphicall the maimum profit (to the nearest thousand dollars) and the output (to the nearest unit) that produces the maimum profit. What is the wholesale price of the radio (to the nearest dollar) at this output? [Compare with Problem 97C.] 99. Construction. A construction compan has 840 feet of chainlink fence that is used to enclose storage areas for equipment and materials at construction sties. The supervisor wants to set up two identical rectangular storage areas sharing a common fence (see the figure). (A) Find a quadratic regression model for the data in Table 1. Estimate the demand at a price level of $180. (B) Find a linear regression model for the data in Table 2. Estimate the suppl at a price level of $180. (C) Does a price level of $180 represent a stable condition, or is the price likel to increase or decrease? Eplain. (D) Use the models in parts (A) and (B) to find the equilibrium point. Write the equilibrium price to the nearest cent and the equilibrium quantit to the nearest unit. 1. Telecommunications. According to the Telecommunications Industr Association, wireless telephone subscriptions grew from about 4 million in 1990 to over 180 million in 200 (Table 3). Let represent ears since TABLE 1 Assuming that all fencing is used, (A) Epress the total area A() enclosed b both pens as a function of. (B) From phsical considerations, what is the domain of the function A? (C) Graph function A in a rectangular coordinate sstem. (D) Use the graph to discuss the number and approimate locations of values of that would produce storage areas with a combined area of 2,000 square feet. (E) Approimate graphicall (to the nearest foot) the values of that would produce storage areas with a combined area of 2,000 square feet. (F) Determine algebraicall the dimensions of the storage areas that have the maimum total combined area. What is the maimum area? 0. Equilibrium point. A compan is planning to introduce a -piece set of nonstick cookware. A marketing compan established price demand and price suppl tables for selected prices (Tables 1 and 2), where is the number of cookware sets people are willing to bu and the compan is willing to sell each month at a price of p dollars per set. Price Demand p D() ($) , , ,22,0 0 TABLE 3 Year Wireless Telephone Subscribers Million Subscribers (A) Find an eponential regression model ( = ab ) for this data. Estimate (to the nearest million) the number of subscribers in 201. (B) Some analsts estimate the number of wireless subscribers in 2018 to be 300 million. How does this compare with the prediction of the model of part (A)? Eplain wh the model will not give reliable predictions far into the future. 2. Medicine. One leukemic cell injected into a health 1 mouse will divide into 2 cells in about 2 da. At the end of the da these 2 cells will divide into 4. This doubling continues until 1 billion cells are formed; then the animal dies with leukemic cells in ever part of the bod. (A) Write an equation that will give the number N of leukemic cells at the end of t das. (B) When, to the nearest da, will the mouse die? TABLE 2 Price Suppl p S() ($) ,14 7 2,90 1 4,22 1, Marine biolog. The intensit of light entering water is reduced according to the eponential equation I = I 0 e -kd where I is the intensit d feet below the surface, I 0 is the intensit at the surface, and k is the coefficient of etinction. Measurements in the Sargasso Sea in the West Indies have indicated that half of the surface light reaches a depth of

81 126 CHAPTER 2 Functions and Graphs 73.6 feet. Find k (to five decimal places), and find the depth (to the nearest foot) at which 1% of the surface light remains. 4. Agriculture. The number of dair cows on farms in the United States is shown in Table 4 for selected ears since 190. Let 1940 be ear 0. TABLE 4 Dair Cows on Farms in the United States Year Dair Cows (thousands) , , , , , ,190 (A) Find a logarithmic regression model ( = a + b ln ) for the data. Estimate (to the nearest thousand) the number of dair cows in 201. (B) Eplain wh it is not a good idea to let 190 be ear 0.. Population growth. Some countries have a relative growth rate of 3% (or more) per ear.at this rate, how man ears (to the nearest tenth of a ear) will it take a population to double? 6. Medicare. The annual ependitures for Medicare (in billions of dollars) b the U.S. government for selected ears since 1980 are shown in Table. Let represent ears since TABLE Medicare Ependitures Year Billion $ (A) Find an eponential regression model ( = ab ) for the data. Estimate (to the nearest billion) the annual ependitures in 201. (B) When will the annual ependitures reach one trillion dollars?