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 21 Functions 22 Elementar Functions: Graphs and Transformations 23 Quadratic Functions 24 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 21 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, = 92 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 pointbpoint plotting. EXAMPLE 1 SOLUTIONS PointbPoint Plotting Sketch the graph of each equation. (A) = 92 (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 = 92
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) = 24 (B) 2 = Eplore & Discuss 1 To graph the equation = , we use pointbpoint 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 pointbpoint 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 pointbpoint 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 gradepoint 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 (11) = 1 # 0 = 0. If = 1, then = 1 (11) =  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) 43 = 8, a real number (B) 22 = 9, a real number (A) Solving for the dependent variable, we have 43 = 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 ( = ), 22 = 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) 24 = 9, a real number (B) 32 = 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 43 = 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 22 = 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 VERTICALLINE 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 A6).
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 verticalline 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 = 22, 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() = 12 h() = then * (A) = 12 f(6) = = 3 (B) g(2) = 1  (2) 2 = 14 = 3 (C) h(2) = = 13 But 13 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() = (112 ) = = 63 = 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() = 122 g() = 12 h() = 21 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() = 122 Z 2 Domain of g: The domain is R, the set of all real numbers, since 12 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 21 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 A2). 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 22a + 7 f(a + h) = (a + h) 22(a + h) + 7 = a 2 + 2ah + h 22a  2h + 7 f(a + h)  f(a) = (a 2 + 2ah + h 22a  2h + 7)  (a 22a + 7) f(a + h)  f(a) h = 2ah + h 22h = 2ah + h22h 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 breakeven 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 24a + 9 (B) a 2 + 2ah + h 24a  4h + 9 (C) 2ah + h 24h (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 pointbpoint 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 pointbpoint 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() = 02 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() = 23 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() = 412 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() = 21, find and simplif each epression in Problems f() 96. f(3) 97. f(2 + ) 98. f(36) 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. Equalsized 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 22 ELEMENTARY FUNCTIONS: GRAPHS AND TRANSFORMATIONS A Beginning Librar of Elementar Functions Vertical and Horizontal Shifts Reflections, Stretches, and Shrinks PiecewiseDefined Functions The functions g() = 24 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 A1). 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 = 14 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 = 14 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() = 24 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 PiecewiseDefined 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 piecewisedefined 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, piecewisedefined functions occur naturall in man applications. (A) 2 2 (B) 2 2 (C) 2 2 if if 0 FIGURE Graphing a piecewisedefined 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 = 14 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 = 14 is the same as the graph of = 1 shifted to the right 4 units. The figure confirms these conclusions H() = , G() = 1 32, 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()
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