1.7 Inverse Functions


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1 71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance, the function f from the set A 1,,, to the set B,, 7, 8 can be written as follows. f : 1,,,,, 7,, 8 In this case, b interchanging the first and second coordinates of each of these ordered pairs, ou can form the inverse function of f, which is denoted b f 1. It is a function from the set B to the set A, and can be written as follows. f 1 :, 1,,, 7,, 8, Note that the domain of f is equal to the range of f 1, and vice versa, as shown in Figure 1.8. Also note that the functions f and f 1 have the effect of undoing each other. In other words, when ou form the composition of f with f 1 or the composition of f 1 with f, ou obtain the identit function. f f 1 f f 1 f f 1 What ou should learn Find inverse functions informall and verif that two functions are inverse functions of each other. Use graphs of functions to decide whether functions have inverse functions. Determine if functions are onetoone. Find inverse functions algebraicall. Wh ou should learn it Inverse functions can be helpful in further eploring how two variables relate to each other. For eample, in Eercises 10 and 10 on page 1, ou will use inverse functions to find the European shoe sizes from the corresponding U.S. shoe sizes. Domain of f f() = + Range of f f() Range of f f () = Domain of f LWADann Tardif/Corbis Figure 1.8 Eample 1 Finding Inverse Functions Informall Find the inverse function of f(). Then verif that both f f 1 and f 1 f are equal to the identit function. The function f multiplies each input b. To undo this function, ou need to divide each input b. So, the inverse function of f is given b f 1. You can verif that both f f 1 and f 1 f are equal to the identit function as follows. f f 1 f Now tr Eercise 1. f 1 f f 1 STUDY TIP Don t be confused b the use of the eponent 1 to denote the inverse function f 1. In this tet, whenever f 1 is written, it alwas refers to the inverse function of the function f and not to the reciprocal of f, which is given b 1 f.
2 71_0107.qd 1/7/0 10: AM Page Chapter 1 Functions and Their Graphs Eample Finding Inverse Functions Informall Find the inverse function of f. Then verif that both f f 1 and f 1 f are equal to the identit function. The function f subtracts from each input. To undo this function, ou need to add to each input. So, the inverse function of f is given b f 1. You can verif that both f f 1 and f 1 f are equal to the identit function as follows. f f 1 f f 1 f f 1 Now tr Eercise. A table of values can help ou understand inverse functions. For instance, the following table shows several values of the function in Eample. Interchange the rows of this table to obtain values of the inverse function f f In the table at the left, each output is less than the input, and in the table at the right, each output is more than the input. The formal definition of an inverse function is as follows. Definition of Inverse Function Let f and g be two functions such that f g for ever in the domain of g and g f for ever in the domain of f. Under these conditions, the function g is the inverse function of the function f. The function g is denoted b f 1 (read finverse ). So, f f 1 and f 1 f. The domain of f must be equal to the range of f 1, and the range of f must be equal to the domain of f 1. If the function g is the inverse function of the function f, it must also be true that the function f is the inverse function of the function g. For this reason, ou can sa that the functions f and g are inverse functions of each other.
3 71_0107.qd 1/7/0 10: AM Page 19 Section 1.7 Inverse Functions 19 Eample Verifing Inverse Functions Algebraicall Show that the functions are inverse functions of each other. f 1 and f g f g f g Eample Now tr Eercise 1. g 1 Verifing Inverse Functions Algebraicall Which of the functions is the inverse function of g or h B forming the composition of f with g, ou have f g f. 1 Because this composition is not equal to the identit function, it follows that g is not the inverse function of f. B forming the composition of f with h, ou have f h f. So, it appears that h is the inverse function of f. You can confirm this b showing that the composition of h with f is also equal to the identit function. Now tr Eercise 19. f? Point out to students that when using a graphing utilit, it is important to know a function s behavior because the graphing utilit ma show an incomplete function. For instance, it is important to know that the domain of f is all real numbers, because a graphing utilit ma show an incomplete graph of the function, depending on how the function was entered. TECHNOLOGY TIP Most graphing utilities can graph 1 in two was: 1 1 or However, ou ma not be able to obtain the complete graph of b entering 1. If not, ou should use 1 1 or = = /
4 71_0107.qd 1/7/0 10: AM Page Chapter 1 Functions and Their Graphs The Graph of an Inverse Function The graphs of a function f and its inverse function f 1 are related to each other in the following wa. If the point a, b lies on the graph of f, then the point b, a must lie on the graph of f 1, and vice versa. This means that the graph of f 1 is a reflection of the graph of f in the line, as shown in Figure 1.8. ( a, b) = f ( ) = = f ( ) TECHNOLOGY TIP In Eamples and, inverse functions were verified algebraicall. A graphing utilit can also be helpful in checking whether one function is the inverse function of another function. Use the Graph Reflection Program found at this tetbook s Online Stud Center to verif Eample graphicall. ( b, a) Figure 1.8 Eample Verifing Inverse Functions Graphicall and Numericall Verif that the functions f and g from Eample are inverse functions of each other graphicall and numericall. Graphical You can verif that f and g are inverse functions of each other graphicall b using a graphing utilit to graph f and g in the same viewing window. (Be sure to use a square setting.) From the graph in Figure 1.8, ou can verif that the graph of g is the reflection of the graph of f in the line. g() = + 1 = f() = 1 Numerical You can verif that f and g are inverse functions of each other numericall. Begin b entering the compositions f g and g f into a graphing utilit as follows. 1 f g 1 1 g f 1 1 Then use the table feature of the graphing utilit to create a table, as shown in Figure 1.8. Note that the entries for, 1, and are the same. So, f g and g f. You can now conclude that f and g are inverse functions of each other. Figure 1.8 Now tr Eercise. Figure 1.8
5 71_0107.qd 1/7/0 10: AM Page 11 Section 1.7 Inverse Functions 11 The Eistence of an Inverse Function Consider the function f. The first table at the right is a table of values for f. The second table was created b interchanging the rows of the first table. The second table does not represent a function because the input is matched with two different outputs: and. So, f does not have an inverse function. To have an inverse function, a function must be onetoone, which means that no two elements in the domain of f correspond to the same element in the range of f. Definition of a OnetoOne Function A function f is onetoone if, for a and b in its domain, f a f b implies that a b f() g() Eistence of an Inverse Function A function f has an inverse function f 1 if and onl if f is onetoone. = From its graph, it is eas to tell whether a function of is onetoone. Simpl check to see that ever horizontal line intersects the graph of the function at most once. This is called the Horizontal Line Test. For instance, Figure 1.8 shows the graph of. On the graph, ou can find a horizontal line that intersects the graph twice. Two special tpes of functions that pass the Horizontal Line Test are those that are increasing or decreasing on their entire domains. 1. If f is increasing on its entire domain, then f is onetoone.. If f is decreasing on its entire domain, then f is onetoone. Eample Testing for OnetoOne Functions Is the function f 1 onetoone? (, 1) 1 1 Figure 1.8 onetoone. (1, 1) f is not Algebraic Let a and b be nonnegative real numbers with f a f b. a 1 b 1 Set f a f b. a b a b So, f a f b implies that a b. You can conclude that f is onetoone and does have an inverse function. Graphical Use a graphing utilit to graph the function 1. From Figure 1.87, ou can see that a horizontal line will intersect the graph at most once and the function is increasing. So, fisonetoone and does have an inverse function. = Now tr Eercise. Figure 1.87
6 71_0107.qd 1/7/0 10: AM Page 1 1 Chapter 1 Functions and Their Graphs Finding Inverse Functions Algebraicall For simple functions, ou can find inverse functions b inspection. For more complicated functions, however, it is best to use the following guidelines. Finding an Inverse Function The function f with an implied domain of all real numbers ma not pass the Horizontal Line Test. In this case, the domain of f ma be restricted so that f does have an inverse function. For instance, if the domain of f is restricted to the nonnegative real numbers, then f does have an inverse function. Eample 7 Finding an Inverse Function Algebraicall Find the inverse function of f 1. Use the Horizontal Line Test to decide whether f has an inverse function.. In the equation for f, replace f b.. Interchange the roles of and, and solve for.. Replace b f 1 in the new equation.. Verif that f and f 1 are inverse functions of each other b showing that the domain of f is equal to the range of f 1, the range of f is equal to the domain of f 1, and f f 1 and f 1 f. The graph of f in Figure 1.88 passes the Horizontal Line Test. So ou know that f is onetoone and has an inverse function. f f 1 Write original function. Replace f b. Interchange and. Multipl each side b. Isolate the term. Solve for. The domains and ranges of f and f f 1 and f 1 f. Replace b f 1. f 1 Now tr Eercise 9.. consist of all real numbers. Verif that TECHNOLOGY TIP Man graphing utilities have a builtin feature for drawing an inverse function. To see how this works, consider the function f. The inverse function of f is given b f 1, 0. Enter the function 1. Then graph it in the standard viewing window and use the draw inverse feature. You should obtain the figure below, which shows both f and its inverse function f 1. For instructions on how to use the draw inverse feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center f () = f () =, f() = f() = Figure 1.88 The draw inverse feature is particularl useful if ou cannot find an epression for the inverse function of a given function. For eample, it would be ver difficult to determine the equation for the inverse function of the onetoone function f However, it is eas to use the technique outlined above to obtain the graph of the inverse function.
7 71_0107.qd 1/7/0 10: AM Page 1 Section 1.7 Inverse Functions 1 Eample 8 Finding an Inverse Function Algebraicall Find the inverse function of f and use a graphing utilit to graph f and f 1 in the same viewing window. f f 1 Write original function. Replace f b. Interchange and. Isolate. Solve for. Replace b f 1. The graph of f in Figure 1.89 passes the Horizontal Line Test. So, ou know that f is onetoone and has an inverse function. The graph of f 1 in Figure 1.89 is the reflection of the graph of f in the line. Verif that f f 1 and f 1 f. Now tr Eercise 1. f () = Figure = f() = Eample 9 Finding an Inverse Function Algebraicall Find the inverse function of f and use a graphing utilit to graph f and f 1 in the same viewing window. f f 1, 0 Write original function. Replace f b. Interchange and. Square each side. Isolate. Solve for. Replace b f 1. The graph of f in Figure 1.90 passes the Horizontal Line Test. So ou know that f is onetoone and has an inverse function. The graph of f 1 in Figure 1.90 is the reflection of the graph of f in the line. Note that the range of f is the interval 0,, which implies that the domain of f 1 is the interval 0,. Moreover, the domain of f is the interval which implies that the range of is the interval,, f 1,. Verif that f f 1 and f 1 f. Now tr Eercise. Activities 1. Given f 7, find f 1. Answer: f 1 7. Show that f and g are inverse functions b showing that f g and g f. f 1 g 1. Describe the graphs of functions that have inverse functions and show how the graph of a function and its inverse function are related. f () = +, 0 (0, (, 0 7 Figure 1.90 ( = ( f() =
8 71_0107.qd 1/7/0 10: AM Page 1 1 Chapter 1 Functions and Their Graphs 1.7 Eercises See for workedout solutions to oddnumbered eercises. Vocabular Check Fill in the blanks. 1. If the composite functions f g and g f, then the function g is the function of f, and is denoted b.. The domain of f is the of f 1, and the of f 1 is the range of f. f 1. The graphs of f and are reflections of each other in the line.. To have an inverse function, a function f must be ; that is, f a f b implies a b.. A graphical test for the eistence of an inverse function is called the Line Test. In Eercises 1 8, find the inverse function of f informall. Verif that f f 1 and f 1 f. 1. f. f 1. f 7. f. f f 8. f In Eercises 9 1, (a) show that f and g are inverse functions algebraicall and (b) use a graphing utilit to create a table of values for each function to numericall show that f and g are inverse functions. 9. f 7, g f 9, f, g 9 g f f 9, 19. f 1, 0. In Eercises 1, match the graph of the function with the graph of its inverse function. [The graphs of the inverse functions are labeled (a), (b), (c), and (d).] (a) (c) f 1 1, 7 0; g 1 0; 9 g 9 g 1, 0 < 1 (b) (d) f, g f 8, f 10, g 8, g In Eercises 1 0, show that f and g are inverse functions algebraicall. Use a graphing utilit to graph f and g in the same viewing window. Describe the relationship between the graphs f, g f 1 g 1, f ; g, 0 9
9 71_0107.qd 1/7/0 10: AM Page 1 Section 1.7 Inverse Functions 1 In Eercises 8, show that f and g are inverse functions (a) graphicall and (b) numericall.. f, f, f 1, f, g g 1 g 1 g 1.. h f In Eercises 7 8, determine algebraicall whether the function is onetoone. Verif our answer graphicall. 7. f 8. g 9. f 0. f In Eercises 9, determine if the graph is that of a function. If so, determine if the function is onetoone f f, q, f f f, 8. f 1 h 1.. In Eercises 9 8, find the inverse function of f algebraicall. Use a graphing utilit to graph both f and f 1 in the same viewing window. Describe the relationship between the graphs f 0. f 1. f. f 1. f. f, 0. f, 0. f 1, 0 In Eercises, use a graphing utilit to graph the function and use the Horizontal Line Test to determine whether the function is onetoone and so has an inverse function eists.. f 1. f h 8. g 1 9. h 1 0. f 1 1. f 10. f 0.. g. f 7 7. f 8. f Think About It In Eercises 9 78, restrict the domain of the function f so that the function is onetoone and has an inverse function. Then find the inverse function f 1. State the domains and ranges of f and f 1. Eplain our results. (There are man correct answers.) 9. f 70. f f 7. f 7. f 7. f 7. f 7. f f f 1
10 71_0107.qd 1//07 1:0 PM Page 1 1 Chapter 1 Functions and Their Graphs In Eercises 79 and 80, use the graph of the function f to complete the table and sketch the graph of f f In Eercises 81 88, use the graphs of f and g to evaluate the function. f = f() = g() 81. f g f g 8. g f 8. f 1 g 0 8. g 1 f 87. g f f 1 g 1 Graphical Reasoning In Eercises 89 9, (a) use a graphing utilit to graph the function, (b) use the draw inverse feature of the graphing utilit to draw the inverse of the function, and (c) determine whether the graph of the inverse relation is an inverse function, eplaining our reasoning. 89. f h 91. g 9. f 1 1 In Eercises 9 98, use the functions f 1 8 and g to find the indicated value or function. 9. f 1 g g 1 f 1 0 f 1 f 1 9. f 1 f 1 9. g 1 g f g g 1 f 1 In Eercises 99 10, use the functions f 1 and g to find the specified function. 99. g 1 f f 1 g f g g f Shoe Sizes The table shows men s shoe sizes in the United States and the corresponding European shoe sizes. Let f represent the function that gives the men s European shoe size in terms of, the men s U.S. size. Men s U.S. shoe size (a) Is f onetoone? Eplain. (b) Find f 11. (c) Find f 1, if possible. (d) Find f f 1 1. (e) Find f 1 f Shoe Sizes The table shows women s shoe sizes in the United States and the corresponding European shoe sizes. Let g epresent the function that gives the women s European shoe size in terms of, the women s U.S. size. (a) Is g onetoone? Eplain. (b) Find g. (c) Find g 1. (d) Find g g 1 9. (e) Find g 1 g. Men s European shoe size Women s U.S. shoe size Women s European shoe size
11 71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions Transportation The total values of new car sales f (in billions of dollars) in the United States from 199 through 00 are shown in the table. The time (in ears) is given b t, with t corresponding to 199. (Source: National Automobile Dealers Association) (a) Does f 1 eist? (b) If f 1 eists, what does it mean in the contet of the problem? (c) If f 1 eists, find f (d) If the table above were etended to 00 and if the total value of new car sales for that ear were $90. billion, would f 1 eist? Eplain. 10. Hourl Wage Your wage is $8.00 per hour plus $0.7 for each unit produced per hour. So, our hourl wage in terms of the number of units produced is (a) Find the inverse function. What does each variable in the inverse function represent? (b) Use a graphing utilit to graph the function and its inverse function. (c) Use the trace feature of a graphing utilit to find the hourl wage when 10 units are produced per hour. (d) Use the trace feature of a graphing utilit to find the number of units produced when our hourl wage is $.. Snthesis Year, t Sales, f t True or False? In Eercises 107 and 108, determine whether the statement is true or false. Justif our answer If f is an even function, f 1 eists If the inverse function of f eists, and the graph of f has a intercept, the intercept of f is an intercept of f Proof Prove that if f and g are onetoone functions, f g 1 g 1 f Proof Prove that if f is a onetoone odd function, f 1 is an odd function. In Eercises , decide whether the two functions shown in the graph appear to be inverse functions of each other. Eplain our reasoning In Eercises , determine if the situation could be represented b a onetoone function. If so, write a statement that describes the inverse function. 11. The number of miles n a marathon runner has completed in terms of the time t in hours 11. The population p of South Carolina in terms of the ear t from 190 to The depth of the tide d at a beach in terms of the time t over a hour period 118. The height h in inches of a human born in the ear 000 in terms of his or her age n in ears Skills Review In Eercises 119 1, write the rational epression in simplest form In Eercises 1 18, determine whether the equation represents as a function of
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