Functions and Their Graphs


 Deirdre McDaniel
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1 3 Functions and Their Graphs On a sales rack of clothes at a department store, ou see a shirt ou like. The original price of the shirt was $00, but it has been discounted 30%. As a preferred shopper, ou get an automatic additional 20% off the sale price at the register. How much will ou pa for the shirt? Naïve shoppers might be lured into thinking this shirt will cost $50 because the add the 20% and 30% to get 50% off, but the will end up paing more than that. Eperienced shoppers know that the first take 30% off of $00, which results in a price of $70, and then the take an additional 20% off of the sale price, $70, which results in a final discounted price of $56. Eperienced shoppers have alread learned composition of functions. A composition of functions can be thought of as a function of a function. One function takes an input (original price, $00) and maps it to an output (sale price, $70), and then another function takes that output as its input (sale price, $70) and maps that to an output (checkout price, $56). John Giustina/Superstock/Photo librar
2 IN THIS CHAPTER ou will find that functions are part of our everda thinking: converting from degrees Celsius to degrees Fahrenheit, DNA testing in forensic science, determining stock values, and the sale price of a shirt. We will develop a more complete, thorough understanding of functions. First, we will establish what a relation is, and then we will determine whether a relation is a function. We will discuss common functions, domain and range of functions, and graphs of functions. We will determine whether a function is increasing or decreasing on an interval and calculate the average rate of change of a function. We will perform operations on functions and composition of functions. We will discuss onetoone functions and inverse functions. Finall, we will model applications with functions using variation. FUNCTIONS AND THEIR GRAPHS 3. Functions 3.2 Graphs of Functions; Piecewise Defined Functions; Increasing and Decreasing Functions; Average Rate of Change 3.3 Graphing Techniques: Transformations 3.4 Operations on Functions and Composition of Functions 3.5 OnetoOne Functions and Inverse Functions 3.6 Modeling Functions Using Variation Relations and Functions Functions Defined b Equations Function Notation Domain of a Function Recognizing and Classifing Functions Increasing and Decreasing Functions Average Rate of Change Piecewise Defined Functions Horizontal and Vertical Shifts Reflection about the Aes Stretching and Compressing Adding, Subtracting, Multipling, and Dividing Functions Composition of Functions Determine Whether a Function Is OnetoOne Inverse Functions Graphical Interpretation of Inverse Functions Finding the Inverse Function Direct Variation Inverse Variation Joint Variation and Combined Variation LEARNING OBJECTIVES Find the domain and range of a function. Sketch the graphs of common functions. Sketch graphs of general functions emploing translations of common functions. Perform composition of functions. Find the inverse of a function. Model applications with functions using variation. 267
3 SECTION 3. FUNCTIONS SKILLS OBJECTIVES Determine whether a relation is a function. Determine whether an equation represents a function. Use function notation. Find the value of a function. Determine the domain and range of a function. CONCEPTUAL OBJECTIVES Think of function notation as a placeholder or mapping. Understand that all functions are relations but not all relations are functions. Relations and Functions What do the following pairs have in common? Ever person has a blood tpe. Temperature is some specific value at a particular time of da. Ever working household phone in the United States has a 0digit phone number. Firstclass postage rates correspond to the weight of a letter. Certain times of the da are start times of sporting events at a universit. The all describe a particular correspondence between two groups. A relation is a correspondence between two sets. The first set is called the domain, and the corresponding second set is called the range. Members of these sets are called elements. D EFINITION Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in the second set, called the range. A relation is a set of ordered pairs. The domain is the set of all the first components of the ordered pairs, and the range is the set of all the second components of the ordered pairs. PERSON BLOOD TYPE ORDERED PAIR Michael A (Michael, A) Tania A (Tania, A) Dlan AB (Dlan, AB) Trevor O (Trevor, O) Megan O (Megan, O) WORDS MATH The domain is the set of all the {Michael, Tania, Dlan, Trevor, Megan} first components. The range is the set of all the {A, AB, O} second components. A relation in which each element in the domain corresponds to eactl one element in the range is a function. 268
4 3. Functions 269 D EFINITION Function A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to eactl one element in the second set, called the range. Note that the definition of a function is more restrictive than the definition of a relation. For a relation, each input corresponds to at least one output, whereas, for a function, each input corresponds to eactl one output. The bloodtpe eample given is both a relation and a function. Also note that the range (set of values to which the elements of the domain correspond) is a subset of the set of all blood tpes. However, although all functions are relations, not all relations are functions. For eample, at a universit, four primar sports TIME OF DAY COMPETITION tpicall overlap in the late fall: football, volleball, soccer, and basketball. On a given Saturda, the following table :00 P.M. Football indicates the start times for the competitions. 2:00 P.M. Volleball Domain PEOPLE Michael Megan Dlan Trevor Tania Function Range BLOOD TYPE A AB O B 7:00 P.M. Soccer 7:00 P.M. Basketball WORDS The :00 start time corresponds to eactl one event, Football. The 2:00 start time corresponds to eactl one event, Volleball. The 7:00 start time corresponds to two events, Soccer and Basketball. MATH (:00 P.M., Football) (2:00 P.M., Volleball) (7:00 P.M., Soccer) (7:00 P.M., Basketball) Domain START TIME :00 P.M. 2:00 P.M. 7:00 P.M. Range Not a Function ATHLETIC EVENT Football Volleball Soccer Basketball Because an element in the domain, 7:00 P.M., corresponds to more than one element in the range, Soccer and Basketball, this is not a function. It is, however, a relation. EXAMPLE Determining Whether a Relation Is a Function Determine whether the following relations are functions. a. {( 3, 4), (2, 4), (3, 5), (6, 4)} b. {( 3, 4), (2, 4), (3, 5), (2, 2)} c. Domain Set of all items for sale in a grocer store; Range Price a. No value is repeated. Therefore, each value corresponds to eactl one value. This relation is a function. b. The value 2 corresponds to both 2 and 4. This relation is not a function. c. Each item in the grocer store corresponds to eactl one price. This relation is a function. YOUR TURN Determine whether the following relations are functions. a. {(, 2), (3, 2), (5, 6), (7, 6)} b. {(, 2), (, 3), (5, 6), (7, 8)} c. {(:00 A.M., 83 F), (2:00 P.M., 89 F), (6:00 P.M., 85 F)} Stud Tip All functions are relations but not all relations are functions. Classroom Eample 3.. Determine whether these relations are functions. a. {(, ), (, ), (0, ), (0, )} b. {(, 2), ( 2, 3), ( 3, 4), ( 4, 5)} c. Domain Set of high school seniors; Range GPA Answer: a. no b. es c. es Answer: a. function b. not a function c. function
5 270 CHAPTER 3 Functions and Their Graphs All of the eamples we have discussed thus far are discrete sets in that the represent a countable set of distinct pairs of (, ). A function can also be defined algebraicall b an equation. Functions Defined b Equations Let s start with the equation 2 3, where can be an real number. This equation assigns to each value eactl one corresponding value. 2 3 () 2 3() 2 5 (5) 2 3(5) = A 2 3 B 23A 2 3 B.2 (.2) 2 3(.2) CAUTION Not all equations are functions. Since the variable depends on what value of is selected, we denote as the dependent variable. The variable can be an number in the domain; therefore, we denote as the independent variable. Although functions are defined b equations, it is important to recognize that not all equations are functions. The requirement for an equation to define a function is that each element in the domain corresponds to eactl one element in the range. Throughout the ensuing discussion, we assume to be the independent variable and to be the dependent variable. Equations that represent functions of : 2 3 Equations that do not represent functions of : Stud Tip We sa that 2 is not a function of. However, if we reverse the independent and dependent variables, then 2 is a function of. In the equations that represent functions of, ever value corresponds to eactl one value. Some ordered pairs that correspond to these functions are 2 : (, ) (0, 0) (, ) : (, ) (0, 0) (, ) 3 : (, ) (0, 0) (, ) The fact that and both correspond to in the first two eamples does not violate the definition of a function. In the equations that do not represent functions of, some values correspond to more than one value. Some ordered pairs that correspond to these equations are RELATION SOLVE RELATION FOR Y POINTS THAT LIE ON THE GRAPH 2 = ; (, ) (0, 0) (, ) maps to both and 2 2 = ;22 (0, ) (0, ) (, 0) (, 0) 0 maps to both and = ; (, ) (0, 0) (, ) maps to both and
6 3. Functions 27 Let s look at the graphs of the three functions of : = 2 = = 3 Let s take an value for, sa a. The graph of a corresponds to a vertical line. A function of maps each value to eactl one value; therefore, there should be at most one point of intersection with an vertical line. We see in the three graphs of the functions above that if a vertical line is drawn at an value of on an of the three graphs, the vertical line onl intersects the graph in one place. Look at the graphs of the three equations that do not represent functions of. = = = A vertical line can be drawn on an of the three graphs such that the vertical line will intersect each of these graphs at two points. Thus, there are two values that correspond to some value in the domain, which is wh these equations do not define as a function of. D EFINITION Vertical Line Test Given the graph of an equation, if an vertical line that can be drawn intersects the graph at no more than one point, the equation defines as a function of. This test is called the vertical line test. Stud Tip If an value corresponds to more than one value, then is not a function of.
7 272 CHAPTER 3 Functions and Their Graphs Classroom Eample 3..2 Use the vertical line test to determine whether these graphs of equations determine functions. a. EXAMPLE 2 Using the Vertical Line Test Use the vertical line test to determine whether the graphs of equations define functions of. a. b. b. Appl the vertical line test. a. b. Answer: a. es b. no Classroom Eample 3..2* Let a be a positive real number. Does the graph of ( a) 2 ( a) 2 4 determine a function? Answer: No, it s a circle. Answer: The graph of the equation is a circle, which does not pass the vertical line test. Therefore, the equation does not define a function. a. Because the vertical line intersects the graph of the equation at two points, this equation does not represent a function. b. Because an vertical line will intersect the graph of this equation at no more than one point, this equation represents a function. YOUR TURN Determine whether the equation ( 3) 2 ( 2) 2 6 is a function of. To recap, a function can be epressed one of four was: verball, numericall, algebraicall, and graphicall. This is sometimes called the Rule of 4. Epressing a Function VERBALLY NUMERICALLY ALGEBRAICALLY GRAPHICALLY Ever real number has a corresponding absolute value. {( 3, 3), (, ), (0, 0), (, ), (5, 5)}
8 3. Functions 273 Function Notation We know that the equation 2 5 defines as a function of because its graph is a nonvertical line and thus passes the vertical line test. We can select values (input) and determine unique corresponding values (output). The output is found b taking 2 times the input and then adding 5. If we give the function a name, sa,, then we can use function notation: f() = The smbol () is read evaluated at or of and represents the value that corresponds to a particular value. In other words, (). INPUT FUNCTION OUTPUT EQUATION () () 2 5 Independent Mapping Dependent Mathematical variable variable rule It is important to note that is the function name, whereas () is the value of the function. In other words, the function maps some value in the domain to some value f() in the range. () 2 5 () 0 (0) 2(0) 5 (0) 5 () 2() 5 () 7 2 (2) 2(2) 5 (2) 9 Domain Function Range f f f f(0) 5 f() 7 f(2) 9 The independent variable is also referred to as the argument of a function. To evaluate functions, it is often useful to think of the independent variable or argument as a placeholder. For eample, () 2 3 can be thought of as f( ) ( ) 2 3( ) In other words, of the argument is equal to the argument squared minus 3 times the argument. An epression can be substituted for the argument: It is important to note: () does not mean f times. f() = () 23() f( + ) = ( + ) 23( + ) f() = () 23() The most common function names are and F since the word function begins with an f. Other common function names are g and G, but an letter can be used. The letter most commonl used for the independent variable is. The letter t is also common because in realworld applications it represents time, but an letter can be used. Although we can think of and () as interchangeable, the function notation is useful when we want to consider two or more functions of the same independent variable. Stud Tip It is important to note that () does not mean times.
9 274 CHAPTER 3 Functions and Their Graphs Classroom Eample 3..3 Let f() 4. Compute: a. f( ) b. 2f() Answer: a. b. 6 EXAMPLE 3 Evaluating Functions b Substitution Given the function () , find ( ). Consider the independent variable to be a placeholder. ( ) 2( ) 3 3( ) 2 6 To find ( ), substitute into the function. ( ) 2( ) 3 3( ) 2 6 Evaluate the right side. ( ) Simplif. ( ) Classroom Eample 3..4 Consider this graph: a. Find f( 3). b. Find f(2). c.* Find such that f() 0. Answer: a. 3 b. c..5, 3 EXAMPLE 4 The graph of is given on the right. a. Find (0). b. Find (). c. Find (2). d. Find 4(3). e. Find such that () 0. f. Find such that () 2. Finding Function Values from the Graph of a Function Solution (a): The value 0 corresponds to the value 5. (0) 5 Solution (b): The value corresponds to the value 2. () 2 Solution (c): The value 2 corresponds to the value. (2) Solution (d): The value 3 corresponds to the value 2. 4(3) Solution (e): The value 0 corresponds to the value 5. Solution (f): The value 2 corresponds to the values and 3. 0 (0, 5) (, 2) (4, 5) (3, 2) (2, ) (5, 0) 5 Answer: a. ( ) 2 b. (0) c. 3(2) 2 d. YOUR TURN For the following graph of a function, find: a. ( ) b. (0) c. 3(2) d. the value of that corresponds to () 0 ( 2, 9) 0 (0, ) (, 2) 5 (, 0) 5 0 (2, 7)
10 3. Functions 275 EXAMPLE 5 C OMMON A common misunderstanding is to interpret the notation ( ) as a sum: f( + ) Z f() + f(). CORRECT Write the original function. f() = 23 M ISTAKE Replace the argument with a placeholder. f ( ) ( ) 2 3( ) Evaluating Functions with Variable Arguments (Inputs) For the given function () 2 3, evaluate ( ) and simplif if possible. INCORRECT The ERROR is in interpreting the notation as a sum. f( + ) Z f() + f() Z Classroom Eample 3..5 Let f() ( 3) 2. Compute: a. f( 3) b. f(3 ) c. f(2 ) Answer: a. 2 b. 2 c CAUTION f( + ) Z f() + f() Substitute for the argument. f( + ) = ( + ) 23( + ) Eliminate the parentheses. f( + ) = Combine like terms. f( + ) = YOUR TURN For the given function g() 2 2 3, evaluate g( ). Answer: g(  ) = EXAMPLE 6 Evaluating Functions: Sums For the given function H() 2 2, evaluate: a. H( ) b. H() H() Solution (a): Classroom Eample 3..6 Let f() 2 2. Compute: a. f( ) b. f() f() Answer: a. 2 2 b. 3 2 Technolog Tip Use a graphing utilit to displa graphs of H( ) ( ) 2 2( ) and 2 H() H() Write the function H in placeholder notation. H( ) ( ) 2 2( ) Substitute for the argument of H. H( ) ( ) 2 2( ) Eliminate the parentheses on the right side. H( ) Combine like terms on the right side. H( + ) = Solution (b): The graphs are not the same. Write H(). H() 2 2 Evaluate H at. H() () 2 2() 3 Evaluate the sum H() H(). H() H() H() + H() = Note: Comparing the results of part (a) and part (b), we see that H( ) H() H().
11 276 CHAPTER 3 Functions and Their Graphs Technolog Tip Use a graphing utilit to displa graphs of G( ) ( ) 2 ( ) and 2 G() ( 2 ). The graphs are not the same. EXAMPLE 7 Evaluating Functions: Negatives For the given function G(t) t 2 t, evaluate: a. G( t) b. G(t) Solution (a): Write the function G in placeholder notation. G( ) ( ) 2 ( ) Substitute t for the argument of G. G( t) ( t) 2 ( t) Eliminate the parentheses on the right side. G(t) = t 2 + t Solution (b): Write G(t). G(t) t 2 t Multipl b. G(t) (t 2 t) Eliminate the parentheses on the right side. G(t) = t 2 + t Note: Comparing the results of part (a) and part (b), we see that G( t) G(t). Classroom Eample 3..7* Let f() Compute f( ). Answer: Classroom Eample 3..8 Let f() ( ) 2. Compute: f(2) a. b. f a 2 f(3) 3 b Answer: a. 0 b. 8 9 CAUTION f a a b b Z f (a) f (b) EXAMPLE 8 Evaluating Functions: Quotients For the given function F() 3 5, evaluate: F() a. Fa b. 2 b F(2) Solution (a): Write F in placeholder notation. F( ) 3( ) 5 Replace the argument with. Simplif the right side. Solution (b): 2 Fa 2 b = 3a 2 b + 5 Fa 2 b = 3 2 Evaluate F(). F() 3() 5 8 Evaluate F(2). F(2) 3(2) 5 Divide F() b F(2). F() F(2) = 8 Note: Comparing the results of part (a) and part (b), we see that F a 2 b F() F(2). Answer: a. G(t 2) 3t 0 b. G(t) G(2) 3t 6 G() c.  G(3) 5 d. Ga 3 b = 3 YOUR TURN Given the function G(t) 3t 4, evaluate: G() a. G(t 2) b. G(t) G(2) c. d. Ga G(3) 3 b Eamples 6, 7, and 8 illustrate the following: f(a + b) Z f(a) + f (b) f (t) Z f (t) f a a b b Z f(a) f(b)
12 3. Functions 277 Now that we have shown that f( + h) Z f() + f(h), we turn our attention to one of the fundamental epressions in calculus: the difference quotient. f( + h)  f() h h Z 0 Eample 9 illustrates the difference quotient, which will be discussed in detail in Section 3.2. For now, we will concentrate on the algebra involved when finding the difference quotient. In Section 3.2, the application of the difference quotient will be the emphasis. EXAMPLE 9 Evaluating the Difference Quotient f( + h)  f() For the function f() = 2 , find, h Z 0. h Use placeholder notation for the function () 2. ( ) ( ) 2 ( ) Calculate ( h). ( h) ( h) 2 ( h) Classroom Eample 3..9 Let f() 2 2. f( + h)  f() Compute, h h Z 0. Answer: ( + 2h + 4) Write the difference quotient. Let ( h) ( h) 2 ( h) and () 2. f( + h)  f() h f( + h) u = C( + h)2  ( + h)d  C 2  D h f( + h)  f() f() r h h Z 0 Eliminate the parentheses inside the first set of brackets. Eliminate the brackets in the numerator. Combine like terms. = [2 + 2h + h h]  [ 2  ] h = 2 + 2h + h h h = 2h + h2  h h Factor the numerator. h(2 + h  ) = h Divide out the common factor, h. 2 h h Z 0 YOUR TURN Evaluate the difference quotient for () 2. Answer: 2 + h Domain of a Function Sometimes the domain of a function is stated eplicitl. For eample, f () 0 domain Here, the eplicit domain is the set of all negative real numbers, (, 0). Ever negative real number in the domain is mapped to a positive real number in the range through the absolute value function. r Domain (, 0) 7 4 f() Range (0, ) 7 4
13 278 CHAPTER 3 Functions and Their Graphs If the epression that defines the function is given but the domain is not stated eplicitl, then the domain is implied. The implicit domain is the largest set of real numbers for which the function is defined and the output value () is a real number. For eample, f() = does not have the domain eplicitl stated. There is, however, an implicit domain. Note that if the argument is negative, that is, if 0, then the result is an imaginar number. In order for the output of the function, (), to be a real number, we must restrict the domain to nonnegative numbers, that is, if 0. FUNCTION f() = IMPLICIT DOMAIN [0, ) In general, we ask the question, what can be? The implicit domain of a function ecludes values that cause a function to be undefined or have outputs that are not real numbers. EXPRESSION THAT DEFINES THE FUNCTION EXCLUDED XVALUES EXAMPLE IMPLICIT DOMAIN Polnomial None () All real numbers Rational values that make the denominator equal to 0 g() = Z;3 or (, 3) (3, 3) (3, ) Radical values that result in a square (even) root of a negative number h() =  5 Ú 5 or [5, ) Technolog Tip To visualize the domain of each function, ask the question: What are the ecluded values in the graph? a. Graph of F() = is shown. The ecluded values are 5 and 5. EXAMPLE 0 Determining the Domain of a Function State the domain of the given functions. 3 a. F() = b. H() = c. G() = Solution (a): Write the original equation. Determine an restrictions on the values of. Solve the restriction equation. State the domain restrictions. F() = 225 Z 0 2 Z 25 or Z ;25 = ;5 Z ; Write the domain in interval notation. (, 5) (5, 5) (5, )
14 3. Functions 279 Solution (b): Write the original equation. Determine an restrictions on the values of Solve the restriction inequalit. State the domain restrictions. Write the domain in interval notation. Solution (c): Write the original equation. Determine an restrictions on the values of. State the domain. Write the domain in interval notation. H() = a, 9 2 d G() = 3  no restrictions R (, ) Classroom Eample 3..0 Find the domain of these functions. a. f() = b. f() = c.* f() =  3 A d.* f() = Answer: a. (, 4) (4, 4) (4, ) b. (, 3] c. (, 0) (0, ) d. (, ) YOUR TURN State the domain of the given functions. a. f() =  3 b. g() = 24 Answer: a. 3 or [3, ) b. Z ;2 or (, 2) (2, 2) (2, ) Applications Functions that are used in applications often have restrictions on the domains due to phsical constraints. For eample, the volume of a cube is given b the function V() 3, where is the length of a side. The function () 3 has no restrictions on, and therefore the domain is the set of all real numbers. However, the volume of an cube has the restriction that the length of a side can never be negative or zero. EXAMPLE Price of Gasoline Following the capture of Saddam Hussein in Iraq in 2003, gas prices in the United States escalated and then finall returned to their precapture prices. Over a 6month period, the average price of a gallon of 87 octane gasoline was given b the function C() , where C is the cost function and represents the number of months after the capture. a. Determine the domain of the cost function. b. What was the average price of gas per gallon 3 months after the capture? Solution (a): Since the cost function C() modeled the price of gas onl for 6 months after the capture, the domain is 0 6 or [0, 6]. Solution (b): Write the cost function. C() Find the value of the function when 3. C(3) 0.05(3) 2 0.3(3).7 Simplif. C(3) = 2.5 The average price per gallon 3 months after the capture was $2.5.
15 280 CHAPTER 3 Functions and Their Graphs EXAMPLE 2 The Dimensions of a Pool Epress the volume of a 30 ft 0 ft rectangular swimming pool as a function of its depth. The volume of an rectangular bo is V lwh, where V is the volume, l is the length, w is the width, and h is the height. In this eample, the length is 30 ft, the width is 0 ft, and the height represents the depth d of the pool. Write the volume as a function of depth d. Simplif. Determine an restrictions on the domain. d 0 V(d) = (30)(0)d V(d) = 300d SECTION 3. SUMMARY Relations and Functions (Let represent the independent variable and the dependent variable.) TYPE MAPPING/CORRESPONDENCE EQUATION GRAPH Relation Ever value in the domain maps to at least one value in the range. 2 Function Ever value in the domain maps to eactl one value in the range. 2 Passes vertical line test All functions are relations, but not all relations are functions. Functions can be represented b equations. In the following table, each column illustrates an alternative notation. INPUT CORRESPONDENCE OUTPUT EQUATION Function 2 5 Independent Mapping Dependent Mathematical variable variable rule Argument () () 2 5 The domain is the set of all inputs (values), and the range is the set of all corresponding outputs (values). Placeholder notation is useful when evaluating functions. f () = f ( ) 3( ) 2 2( ) Eplicit domain is stated, whereas implicit domain is found b ecluding values that: make the function undefined (denominator 0). result in a nonreal output (even roots of negative real numbers).
16 3. Functions 28 SECTION 3. EXERCISES SKILLS In Eercises 24, determine whether each relation is a function. Assume that the coordinate pair (, ) represents the independent variable and the dependent variable Domain Range Domain Range Domain Range MONTH October Januar April AVERAGE TEMPERATURE 78 F 68 F PERSON Mar Jason Chester 0DIGIT PHONE # (202) (307) (878) START TIME :00 P.M. 4:00 P.M. 7:00 P.M. NFL GAME Bucs/Panthers Bears/Lions Falcons/Saints Rams/Seahawks Packers/Vikings Domain Range Domain Range Domain Range PERSON Jordan Pat DATE THIS WEEKEND Chris Ale Morgan PERSON Carrie Michael Jennifer Sean COURSE GRADE A B MATH SAT SCORE PERSON Carrie Michael Jennifer Sean 7. {(0, 3), (0, 3), ( 3, 0), (3, 0)} 8. {(2, 2), (2, 2), (5, 5), (5, 5)} 9. {(0, 0), (9, 3), (4, 2), (4, 2), (9, 3)} 0. {(0, 0), (, ), ( 2, 8), (, ), (2, 8)}. {(0, ), (, 0), (2, ), ( 2, ), (5, 4), ( 3, 4)} 2. {(0, ), (, ), (2, ), (3, )} (0, 5) ( 5, 0) (5, 0) (0, 5)
17 282 CHAPTER 3 Functions and Their Graphs In Eercises 25 32, use the given graphs to evaluate the functions. 25. () 26. g() 27. p() 28. r() 5 (2, 5) ( 5, 7) (, 3) ( 3, 5) (, 5) (, 4) ( 3, ) (0, ) (5, 0) (, 3) ( 4, 0) 5 (, ) (0, 2) 0 0 (3, 2) (7, 3) 7 3 ( 6, 6) ( 2, 3) (3, 5) 5 a. (2) b. (0) c. ( 2) 5 3 (0, 5) a. g( 3) b. g(0) c. g(5) a. p( ) b. p(0) c. p() 0 a. r( 4) b. r( ) c. r(3) 29. C() 30. q() 3. S() 32. T() 5 5 ( 4, 5) (4, 5) 4 (6, 0) (6, ) ( 3, 4) (4, 3) (5, ) (2, 4) 8 (2, 5) 5 a. C(2) b. C(0) c. C( 2) a. q( 4) b. q(0) c. q(2) a. S( 3) b. S(0) c. S(2) a. T( 5) b. T( 2) c. T(4) 33. Find if f() 3 in Eercise Find if g() 2 in Eercise Find if p() 5 in Eercise Find if C() 7 in Eercise Find if C() 5 in Eercise Find if q() 2 in Eercise Find if S() in Eercise Find if T() 4 in Eercise 32. In Eercises 4 56, evaluate the given quantities appling the following four functions. () 2 3 F(t) 4 t 2 g(t) 5 t G() ( 2) 42. G( 3) 43. g() 44. F( ) 45. ( 2) g() 46. G( 3) F( ) 47. 3f( 2) 2g() 48. 2F( ) 2G( 3) 49. f(2) G(3) f(0)  f(2) G(0)  G(3) g() F() g() F() 53. ( ) ( ) 54. F(t ) F(t ) 55. g( a) ( a) 56. G( b) F(b) In Eercises 57 64, evaluate the difference quotients using the same, F, G, and g given for Eercises f( + h)  f() F(t + h)  F(t) g(t + h)  g(t) h h h f(2 + h)  f(2) F( + h)  F() g( + h)  g() h h h 64. G( + h)  G() h G(3 + h)  G(3) h
18 3. Functions 283 In Eercises 65 96, find the domain of the given function. Epress the domain in interval notation. 65. () f() g(t) t 2 3t 68. h() P() = Q(t) = 2  t2 7. T() = t F() = 74. G(t) = 75. q() = t f() = g() = G(t) = 2t R() = 2  k(t) = t  7 F() = F() = 82. G() = 83. f() = g() = P() = Q() = 87. R() = 88. p() = H(t) = t 90. f(t) = t () ( 2 6) /2 92. g() (2 5) /3 2t 2  t t r() 2 (3 2) /2 94. p() ( ) 2 ( 2 9) 3/5 95. f() = g() = Let g() and find the values of that correspond to g() Let g() = and find the value of that corresponds to g() = Let () 2( 5) 3 2( 5) 2 and find the values of that correspond to () Let () 3( 3) 2 6( 3) 3 and find the values of that correspond to () 0. APPLICATIONS 0. Budget: Event Planning. The cost associated with a catered wedding reception is $45 per person for a reception for more than 75 people. Write the cost of the reception in terms of the number of guests and state an domain restrictions. 02. Budget: LongDistance Calling. The cost of a local home phone plan is $35 for basic service and $.0 per minute for an domestic longdistance calls. Write the cost of monthl phone service in terms of the number of monthl longdistance minutes and state an domain restrictions. 03. Temperature. The average temperature in Tampa, Florida, in the springtime is given b the function T() , where T is the temperature in degrees Fahrenheit and is the time of da in militar time and is restricted to 6 8 (sunrise to sunset). What is the temperature at 6 A.M.? What is the temperature at noon? 04. Falling Objects: Firecrackers. A firecracker is launched straight up, and its height is a function of time, h(t) 6t 2 28t, where h is the height in feet and t is the time in seconds with t 0 corresponding to the instant it launches. What is the height 4 seconds after launch? What is the domain of this function? 05. Collectibles. The price of a signed Ale Rodriguez baseball card is a function of how man are for sale. When Rodriguez was traded from the Teas Rangers to the New York Yankees in 2004, the going rate for a signed baseball card on eba was P() = ,00000, where represents the number of signed cards for sale. What was the value of the card when there were 0 signed cards for sale? What was the value of the card when there were 00 signed cards for sale?
19 284 CHAPTER 3 Functions and Their Graphs 06. Collectibles. In Eercise 05, what was the lowest price on eba, and how man cards were available then? What was the highest price on eba, and how man cards were available then? 07. Volume. An open bo is constructed from a square 0inch piece of cardboard b cutting squares of length inches out of each corner and folding the sides up. Epress the volume of the bo as a function of, and state the domain. 08. Volume. A clindrical water basin will be built to harvest rainwater. The basin is limited in that the largest radius it can have is 0 feet. Write a function representing the volume of water V as a function of height h. How man additional gallons of water will be collected if ou increase the height b 2 feet? Hint: cubic foot 7.48 gallons. For Eercises 09 0, refer to the following: The weekl echange rate of the U.S. dollar to the Japanese en is shown in the graph as varing over an 8week period. Assume the echange rate E(t) is a function of time (week); let E() be the echange rate during Week. Japanese Yen to One U.S. Dollar E Week 09. Economics. Approimate the echange rates of the U.S. dollar to the nearest en during Weeks 4, 7, and Economics. Find the increase or decrease in the number of Japanese en to the U.S. dollar echange rate, to the nearest en, from (a) Week 2 to Week 3 and (b) Week 6 to Week 7. For Eercises 2, refer to the following: An epidemiological stud of the spread of malaria in a rural area finds that the total number P of people who contracted malaria t das into an outbreak is modeled b the function P(t) =  4 t 2 + 7t + 80 t 4. Medicine/Health. How man people have contracted malaria 4 das into the outbreak? 2. Medicine/Health. How man people have contracted malaria 6 das into the outbreak? t 3. Environment: Tossing the Envelopes. The average American adult receives 24 pieces of mail per week, usuall of some combination of ads and envelopes with windows. Suppose each of these adults throws awa a dozen envelopes per week. a. The width of the window of an envelope is inches less than its length. Create the function A() that represents the area of the window in square inches. Simplif, if possible. b. Evaluate A(4.5) and eplain what this value represents. c. Assume the dimensions of the envelope are 8 inches b 4 inches. Evaluate A(8.5). Is this possible for this particular envelope? Eplain. 4. Environment: Tossing the Envelopes. Each month, Jack receives his bank statement in a 9.5 inch b 6 inch envelope. Each month, he throws awa the envelope after removing the statement. a. The width of the window of the envelope is inches less than its length. Create the function A() that represents the area of the window in square inches. Simplif, if possible. b. Evaluate A(5.25) and eplain what this value represents. c. Evaluate A(0). Is this possible for this particular envelope? Eplain. Refer to the table below for Eercises 5 and 6. It illustrates the average federal funds rate for the month of Januar (2000 to 2008). YEAR FED. RATE Finance. Is the relation whose domain is the ear and whose range is the average federal funds rate for the month of Januar a function? Eplain. 6. Finance. Write five ordered pairs whose domain is the set of even ears from 2000 to 2008 and whose range is the set of corresponding average federal funds rate for the month of Januar.
20 3. Functions 285 For Eercises 7 and 8, use the following figure: EmploerProvided Health Insurance Premiums for Famil Plans ( , adjusted for inflation) YEAR $2,000 0,000 8,000 6,000 4,000 2,000 Emploee Share Emploer Share Year Source: Kaiser Famil Foundation Health Research and Education Trust. Note: The following ears were interpolated: ; ; Health Care Costs. Fill in the following table. Round dollars to the nearest $000. TOTAL HEALTH CARE COST FOR FAMILY PLANS Write the five ordered pairs resulting from the table. 8. Health Care Costs. Using the table found in Eercise 7, let the ears correspond to the domain and the total costs correspond to the range. Is this relation a function? Eplain. For Eercises 9 and 20, use the following information: Metric Tons of Carbon/Year (in millions) 7,000 6,000 5,000 4,000 3,000 2,000,000 Global Fossil Carbon Emissions Total Petroleum Coal Natural Gas Cement Production Year Source: Let the functions f, F, g, G, and H represent the number of tons of carbon emitted per ear as a function of ear corresponding to cement production, natural gas, coal, petroleum, and the total amount, respectivel. Let t represent the ear, with t 0 corresponding to Environment: Global Climate Change. Estimate (to the nearest thousand) the value of a. F(50) b. g(50) c. H(50) 20. Environment: Global Climate Change. Eplain what the sum F(00) + g(00) + G(00) represents. CATCH THE MISTAKE In Eercises 2 26, eplain the mistake that is made. 2. Determine whether the relationship is a function. 22. Given the function H() 3 2, evaluate the quantit H(3) H( ). H(3) H( ) H(3) H() 7 8 Appl the horizontal line test. Because the horizontal line intersects the graph in two places, this is not a function. This is incorrect. What mistake was made? This is incorrect. What mistake was made? 23. Given the function () 2, evaluate the quantit ( ). ( ) () () 2 0 ( ) 2 This is incorrect. What mistake was made? 24. Determine the domain of the function g(t) = 3  t and epress it in interval notation. What can t be? An nonnegative real number. 3 t 0 3 t or t 3 Domain: (, 3) This is incorrect. What mistake was made?
21 286 CHAPTER 3 Functions and Their Graphs 25. Given the function G() 2, evaluate G( + h)  G(). h G( + h)  G() G() + G(h)  G() = h h = G(h) h = h2 h = h This is incorrect. What mistake was made? 26. Given the functions () A and (), find A. Since (), the point (, ) must satisf the function. A Add to both sides of the equation. A 0 The absolute value of zero is zero, so there is no need for the absolute value signs:   A = 0 A = . This is incorrect. What mistake was made? CONCEPTUAL In Eercises 27 30, determine whether each statement is true or false. 27. If a vertical line does not intersect the graph of an equation, 30. If ( a) (a), then ma or ma not represent a function. then that equation does not represent a function. 3. If () A 2 3 and (), find A. 28. If a horizontal line intersects a graph of an equation more than once, the equation does not represent a function. 32. If g() = and g(3) is undefined, find b. b If ( a) (a), then does not represent a function. CHALLENGE 33. If F() = C  is undefined, and F( ) 4, find D , F(2) C and D. 34. Construct a function that is undefined at 5 and whose graph passes through the point (, ). In Eercises 35 and 36, find the domain of each function, where a is an positive real number. 35. f() = a f() = a 2 TECHNOLOGY 37. Using a graphing utilit, graph the temperature function in Eercise 03. What time of da is it the warmest? What is the temperature? Looking at this function, eplain wh this model for Tampa, Florida, is valid onl from sunrise to sunset (6 to 8). 38. Using a graphing utilit, graph the height of the firecracker in Eercise 04. How long after liftoff is the firecracker airborne? What is the maimum height that the firecracker attains? Eplain wh this height model is valid onl for the first 8 seconds. 39. Using a graphing utilit, graph the price function in Eercise 05. What are the lowest and highest prices of the cards? Does this agree with what ou found in Eercise 06? 40. The makers of malted milk balls are considering increasing the size of the spherical treats. The thin chocolate coating on a malted milk ball can be approimated b the surface area, S(r) 4 r 2. If the radius is increased 3 mm, what is the resulting increase in required chocolate for the thin outer coating? 4. Let f() = 2 +. Graph = f() and 2 = f(  2) in the same viewing window. Describe how the graph of 2 can be obtained from the graph of. 42. Let f() = 42. Graph = f() and 2 = f( + 2) in the same viewing window. Describe how the graph of 2 can be obtained from the graph of.
22 SECTION 3.2 GRAPHS OF FUNCTIONS; PIECEWISEDEFINED FUNCTIONS; INCREASING AND DECREASING FUNCTIONS; AVERAGE RATE OF CHANGE SKILLS OBJECTIVES Classif functions as even, odd, or neither. Determine whether functions are increasing, decreasing, or constant. Calculate the average rate of change of a function. Evaluate the difference quotient for a function. Graph piecewisedefined functions. CONCEPTUAL OBJECTIVES Identif common functions. Develop and graph piecewisedefined functions. Identif and graph points of discontinuit. State the domain and range. Understand that even functions have graphs that are smmetric about the ais. Understand that odd functions have graphs that are smmetric about the origin. Recognizing and Classifing Functions Common Functions Pointplotting techniques were introduced in Section 2.2, and we noted there that we would eplore some more efficient was of graphing functions in Chapter 3. The nine main functions ou will read about in this section will constitute a librar of functions that ou should commit to memor. We will draw on this librar of functions in the net section when graphing transformations are discussed. Several of these functions have been shown previousl in this chapter, but now we will classif them specificall b name and identif properties that each function ehibits. In Section 2.3, we discussed equations and graphs of lines. All lines (with the eception of vertical lines) pass the vertical line test, and hence are classified as functions. Instead of the traditional notation of a line, m b, we use function notation and classif a function whose graph is a line as a linear function. LINEAR FUNCTION f() = m + b m and b are real numbers. The domain of a linear function () m b is the set of all real numbers R. The graph of this function has slope m and intercept b. LINEAR FUNCTION: f() m b SLOPE: m INTERCEPT: b () 2 7 m 2 b 7 () 3 m b 3 () m b 0 () 5 m 0 b 5 287
23 288 CHAPTER 3 Functions and Their Graphs One special case of the linear function is the constant function (m 0). CONSTANT FUNCTION f() = b b is an real number. The graph of a constant function () b is a horizontal line. The intercept corresponds to the point (0, b). The domain of a constant function is the set of all real numbers R. The range, however, is a single value b. In other words, all values correspond to a single value. Points that lie on the graph of a constant function () b are Domain: (, ) Range: [b, b] or {b} ( 5, b) (, b) (0, b) ( 5, b) (0, b) (4, b) (2, b) (4, b)... (, b) Identit Function Domain: (, ) Range: (, ) Another specific eample of a linear function is the function having a slope of one (m ) and a intercept of zero (b 0). This special case is called the identit function. (3, 3) I DENTITY FUNCTION (2, 2) (0, 0) ( 2, 2) ( 3, 3) () The graph of the identit function has the following properties: It passes through the origin, and ever point that lies on the line has equal  and coordinates. Both the domain and the range of the identit function are the set of all real numbers R. A function that squares the input is called the square function. Square Function Domain: (, ) Range: [0, ) SQUARE FUNCTION f() = 2 ( 2, 4) (, ) (, ) (2, 4) The graph of the square function is called a parabola and will be discussed in further detail in Chapters 4 and 8. The domain of the square function is the set of all real numbers R. Because squaring a real number alwas ields a positive number or zero, the range of the square function is the set of all nonnegative numbers. Note that the intercept is the origin and the square function is smmetric about the ais. This graph is contained in quadrants I and II.
24 3.2 Graphs of Functions 289 A function that cubes the input is called the cube function. CUBE FUNCTION f() = 3 Cube Function Domain: (, ) Range: (, ) 0 (2, 8) The domain of the cube function is the set of all real numbers R. Because cubing a negative number ields a negative number, cubing a positive number ields a positive number, and cubing 0 ields 0, the range of the cube function is also the set of all real numbers R. Note that the onl intercept is the origin and the cube function is smmetric about the origin. This graph etends onl into quadrants I and III. The net two functions are counterparts of the previous two functions: square root and cube root. When a function takes the square root of the input or the cube root of the input, the function is called the square root function or the cube root function, respectivel. SQUARE ROOT FUNCTION f () = or f () = /2 5 5 ( 2, 8) 0 Square Root Function Domain: [0, ) Range: [0, ) 5 (9, 3) (4, 2) In Section 3., we found the domain to be [0, ). The output of the function will be all real numbers greater than or equal to zero. Therefore, the range of the square root function is [0, ). The graph of this function will be contained in quadrant I. CUBE ROOT FUNCTION In Section 3., we stated the domain of the cube root function to be (, ). We see b the graph that the range is also (, ). This graph is contained in quadrants I and III and passes through the origin. This function is smmetric about the origin. In Section.7, ou read about absolute value equations and inequalities. Now we shift our focus to the graph of the absolute value function. ABSOLUTE VALUE FUNCTION f () = 3 or f () = /3 0 Cube Root Function Domain: (, ) Range: (, ) 5 (8, 2) 0 ( 8, 2) Absolute Value Function Domain: (, ) Range: [0, ) f() = Some points that are on the graph of the absolute value function are (, ), (0, 0), and (, ). The domain of the absolute value function is the set of all real numbers R, et the range is the set of nonnegative real numbers. The graph of this function is smmetric with respect to the ais and is contained in quadrants I and II. ( 2, 2) (2, 2)
25 290 CHAPTER 3 Functions and Their Graphs Reciprocal Function Domain: (, 0) (0, ) Range: (, 0) (0, ) A function whose output is the reciprocal of its input is called the reciprocal function. R ECIPROCAL FUNCTION f() = f() = Z 0 (, ) (, ) The onl restriction on the domain of the reciprocal function is that Z 0. Therefore, we sa the domain is the set of all real numbers ecluding zero. The graph of the reciprocal function illustrates that its range is also the set of all real numbers ecept zero. Note that the reciprocal function is smmetric with respect to the origin and is contained in quadrants I and III. Even and Odd Functions Of the nine functions discussed above, several have similar properties of smmetr. The constant function, square function, and absolute value function are all smmetric with respect to the ais. The identit function, cube function, cube root function, and reciprocal function are all smmetric with respect to the origin. The term even is used to describe functions that are smmetric with respect to the ais, or vertical ais, and the term odd is used to describe functions that are smmetric with respect to the origin. Recall from Section 2.2 that smmetr can be determined both graphicall and algebraicall. The bo below summarizes the graphic and algebraic characteristics of even and odd functions. EVEN AND ODD FUNCTIONS Function Smmetric with Respect to On Replacing with Even ais or vertical ais ( ) () Odd origin ( ) () The algebraic method for determining smmetr with respect to the ais, or vertical ais, is to substitute for. If the result is an equivalent equation, the function is smmetric with respect to the ais. Some eamples of even functions are () b, () 2, () 4 ; and (). In an of these equations, if is substituted for, the result is the same; that is, ( ) (). Also note that, with the eception of the absolute value function, these eamples are all evendegree polnomial equations. All constant functions are degree zero and are even functions. The algebraic method for determining smmetr with respect to the origin is to substitute for. If the result is the negative of the original function, that is, if ( ) (), then the function is smmetric with respect to the origin and, hence, classified as an odd function. Eamples of odd functions are (), () 3, () 5, and () /3. In an of these functions, if is substituted for, the result is the negative of the original function. Note that with the eception of the cube root function, these equations are odddegree polnomials.
26 3.2 Graphs of Functions 29 Be careful, though, because functions that are combinations of even and odddegree polnomials can turn out to be neither even nor odd, as we will see in Eample. EXAMPLE Determining Whether a Function Is Even, Odd, or Neither Determine whether the functions are even, odd, or neither. a. () 2 3 b. g() 5 3 c. h() 2 Solution (a): Original function. () 2 3 Replace with. ( ) ( ) 2 3 Simplif. ( ) 2 3 () Because ( ) (), we sa that () is an even function. Solution (b): Original function. g() 5 3 Replace with. g( ) ( ) 5 ( ) 3 Simplif. g( ) 5 3 ( 5 3 ) g() Because g( ) g(), we sa that g() is an odd function. Solution (c): Original function. Replace with. Simplif. h() 2 h( ) ( ) 2 ( ) h( ) 2 Classroom Eample 3.2. Determine whether these functions are even, odd, or neither. a. f() = b. f () = c.* f () = ( ) d. f() = e.* f() = ( ) 2 Answer: a. odd b. odd c. odd d. neither e. even h( ) is neither h() nor h(); therefore the function h() is neither even nor odd. In parts (a), (b), and (c), we classified these functions as either even, odd, or neither, using the algebraic test. Look back at them now and reflect on whether these classifications agree with our intuition. In part (a), we combined two functions: the square function and the constant function. Both of these functions are even, and adding even functions ields another even function. In part (b), we combined two odd functions: the fifthpower function and the cube function. Both of these functions are odd, and adding two odd functions ields another odd function. In part (c), we combined two functions: the square function and the identit function. The square function is even, and the identit function is odd. In this part, combining an even function with an odd function ields a function that is neither even nor odd and, hence, has no smmetr with respect to the vertical ais or the origin. Technolog Tip a. Graph = f() = 23. Even; smmetric with respect to the ais. b. Graph = g() = Odd; smmetric with respect to origin. c. Graph = h() = 2 . No smmetr with respect to ais or origin. YOUR TURN Classif the functions as even, odd, or neither. a. () 4 b. () 3 Answer: a. even b. neither
27 292 CHAPTER 3 Functions and Their Graphs (, ) Stud Tip (0, ) ( 2, 2) (2, 2) (6, 4) Graphs are read from left to right. Intervals correspond to the coordinates. Increasing and Decreasing Functions Look at the figure in the margin to the left. Graphs are read from left to right. If we start at the left side of the graph and trace the red curve with our pen, we see that the function values (values in the vertical direction) are decreasing until arriving at the point ( 2, 2). Then, the function values increase until arriving at the point (, ). The values then remain constant ( ) between the points (, ) and (0, ). Proceeding beond the point (0, ), the function values decrease again until the point (2, 2). Beond the point (2, 2), the function values increase again until the point (6, 4). Finall, the function values decrease and continue to do so. When specifing a function as increasing, decreasing, or constant, the intervals are classified according to the coordinate. For instance, in this graph, we sa the function is increasing when is between 2 and and again when is between 2 and 6. The graph is classified as decreasing when is less than 2 and again when is between 0 and 2 and again when is greater than 6. The graph is classified as constant when is between and 0. In interval notation, this is summarized as Decreasing Increasing Constant (, 2) (0, 2) (6, ) (2, ) (2, 6) (, 0) An algebraic test for determining whether a function is increasing, decreasing, or constant is to compare the value () of the function for particular points in the intervals. I NCREASING, DECREASING, AND CONSTANT FUNCTIONS. A function is increasing on an open interval I if for an and 2 in I, where 2, then ( ) ( 2 ). 2. A function is decreasing on an open interval I if for an and 2 in I, where 2, then ( ) ( 2 ). 3. A function f is constant on an open interval I if for an and 2 in I, then ( ) ( 2 ). In addition to classifing a function as increasing, decreasing, or constant, we can also determine the domain and range of a function b inspecting its graph from left to right: The domain is the set of all values (from left to right) where the function is defined. The range is the set of all values (from bottom to top) that the graph of the function corresponds to. A solid dot on the left or right end of a graph indicates that the graph terminates there and the point is included in the graph. An open dot indicates that the graph terminates there and the point is not included in the graph. Unless a dot is present, it is assumed that a graph continues indefinitel in the same direction. (An arrow is used in some books to indicate direction.)
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