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333371 0107.qxd12/27/0610:35 AMPage 147Section 1.7Inverse Functions1471.7 Inverse FunctionsWhat you should learnInverse FunctionsRecall from Section 1.3 that a function can be represented by a set of orderedpairs. For instance, the function f x x 4 from the set A 1, 2, 3, 4 to theset B 5, 6, 7, 8 can be written as follows.f x x 4: 1, 5 , 2, 6 , 3, 7 , 4, 8 䊏䊏䊏In this case, by interchanging the first and second coordinates of each of theseordered pairs, you can form the inverse function of f, which is denoted by f 1.It is a function from the set B to the set A, and can be written as follows.f 1 x x 4: 5, 1 , 6, 2 , 7, 3 , 8, 4 Note that the domain of f is equal to the range of f 1, and vice versa, as shownin Figure 1.82. Also note that the functions f and f 1 have the effect of “undoing”each other. In other words, when you form the composition of f with f 1 or thecomposition of f 1 with f, you obtain the identity function.䊏Find inverse functions informally andverify that two functions are inversefunctions of each other.Use graphs of functions to decide whetherfunctions have inverse functions.Determine if functions are one-to-one.Find inverse functions algebraically.Why you should learn itInverse functions can be helpful in furtherexploring how two variables relate to eachother. For example, in Exercises 103 and 104on page 156, you will use inverse functionsto find the European shoe sizes from thecorresponding U.S. shoe sizes.f f 1 x f x 4 x 4 4 xf 1 f x f 1 x 4 x 4 4 xf (x) x 4Domain of fRange of fxf(x)Range of f 1f 1Domain of f 1LWA-Dann Tardif/Corbis(x) x 4Figure 1.82Example 1 Finding Inverse Functions InformallyFind the inverse function of f(x) 4x. Then verify that both f f 1 x andf 1 f x are equal to the identity function.SolutionThe function f multiplies each input by 4. To “undo” this function, you need todivide each input by 4. So, the inverse function of f x 4x is given byxf 1 x .4You can verify that both f f 1 x and f 1 f x are equal to the identity functionas follows.f f 1 x f 4 4 4 xxxNow try Exercise 1.f 1 f x f 1 4x 4x x4STUDY TIPDon’t be confused by the useof the exponent 1 to denotethe inverse function f 1. In thistext, whenever f 1 is written,it always refers to the inversefunction of the function f andnot to the reciprocal of f x ,which is given by1.f x

333371 0107.qxd12/27/0614810:35 AMChapter 1Page 148Functions and Their GraphsExample 2 Finding Inverse Functions InformallyFind the inverse function of f x x 6. Then verify that both f f 1 x andf 1 f x are equal to the identity function.SolutionThe function f subtracts 6 from each input. To “undo” this function, you need toadd 6 to each input. So, the inverse function of f x x 6 is given byf 1 x x 6.You can verify that both f f 1 x and f 1 f x are equal to the identity functionas follows.f f 1 x f x 6 x 6 6 xf 1 f x f 1 x 6 x 6 6 xNow try Exercise 3.A table of values can help you understand inverse functions. For instance, thefollowing table shows several values of the function in Example 2. Interchangethe rows of this table to obtain values of the inverse function.x 2 1012f x 8 7 6 5 4x 8 7 6 5 4f 1 x 2 1012In the table at the left, each output is 6 less than the input, and in the table at theright, each output is 6 more than the input.The formal definition of an inverse function is as follows.Definition of Inverse FunctionLet f and g be two functions such thatf g x xfor every x in the domain of gg f x xfor every x in the domain of f.andUnder these conditions, the function g is the inverse function of thefunction f. The function g is denoted by f 1 (read “f-inverse”). So,f f 1 x xandf 1 f x x.The domain of f must be equal to the range of f 1, and the range of f mustbe equal to the domain of f 1.If the function g is the inverse function of the function f, it must also be truethat the function f is the inverse function of the function g. For this reason, youcan say that the functions f and g are inverse functions of each other.

333371 0107.qxd12/27/0610:35 AMPage 149Section 1.7149Inverse FunctionsExample 3 Verifying Inverse Functions AlgebraicallyShow that the functions are inverse functions of each other.f x 2x3 1g x and x 2 13Point out to students that when using agraphing utility, it is important to know afunction’s behavior because the graphingutility may show an incomplete function.For instance, it is important to know that thedomain of f x x 2 3 is all real numbers,because a graphing utility may show anincomplete graph of the function, dependingon how the function was entered.Solutionf g x f x 2 1 2 x 2 1 1333 2 x 2 1 1 x 1 1 xg f x g 2x3 1 2x33 3 x3 3 1 12TECHNOLOGY TIPMost graphing utilities can graphy x1 3 in two ways:2x 323y1 x. xNow try Exercise 15.Example 4 Verifying Inverse Functions AlgebraicallyWhich of the functions is the inverse function of f x g x x 25h x or 1 3 ory1 x5?x 2However, you may not be ableto obtain the complete graphof y x2 3 by enteringy1 x 2 3 . If not, youshould usey1 x 1 3 2 or3 x2 .y1 5 2x5y x2/3SolutionBy forming the composition of f with g, you havef g x f 65x 225 x.5x 2x 12 25 3 y Because this composition is not equal to the identity function x, it follows that gis not the inverse function of f. By forming the composition of f with h, you havef h x f6 x 2 55 5 x.5x x 2 2 5So, it appears that h is the inverse function of f. You can confirm this by showingthat the composition of h with f is also equal to the identity function.Now try Exercise 19.3x25 66 3

333371 0107.qxd15012/27/0610:35 AMChapter 1Page 150Functions and Their GraphsThe Graph of an Inverse FunctionTECHNOLOGY TIPThe graphs of a function f and its inverse function f 1 are related to each other inthe following way. 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 isa reflection of the graph of f in the line y x, as shown in Figure 1.83.yIn Examples 3 and 4, inversefunctions were verifiedalgebraically. A graphing utilitycan also be helpful in checkingwhether one function is theinverse function of anotherfunction. Use the GraphReflection Program found atthis textbook’s Online StudyCenter to verify Example 4graphically.y xy f ( x)(a , b)y f 1 (x)(b , a )xFigure 1.83Example 5 Verifying Inverse Functions Graphicallyand NumericallyVerify that the functions f and g from Example 3 are inverse functions of eachother graphically and numerically.Graphical SolutionNumerical SolutionYou can verify that f and g are inverse functions of each othergraphically by using a graphing utility to graph f and g in thesame viewing window. (Be sure to use a square setting.)From the graph in Figure 1.84, you can verify that the graphof g is the reflection of the graph of f in the line y x.You can verify that f and g are inverse functions of eachother numerically. Begin by entering the compositionsf g x and g f x into a graphing utility as follows.g(x) 3x 124y x 6y2 g f x 6 4y1 f g x 2f(x) 2x3 1 3x 12 2x33 1 12Then use the table feature of the graphing utility tocreate a table, as shown in Figure 1.85. Note that theentries for x, y1, and y2 are the same. So, f g x x andg f x x. You can now conclude that f and g areinverse functions of each other.Figure 1.84Now try Exercise 25. 13Figure 1.85

333371 0107.qxd12/27/0610:35 AMPage 151Section 1.7151Inverse FunctionsThe Existence of an Inverse FunctionConsider the function f x x2. The first table at the right is a table of values forf x x2. The second table was created by interchanging the rows of the firsttable. The second table does not represent a function because the input x 4 ismatched with two different outputs: y 2 and y 2. So, f x x2 does nothave an inverse function.To have an inverse function, a function must be one-to-one, which meansthat no two elements in the domain of f correspond to the same element in therange of f. 2 1012f(x)41014x41014 2 1012xg(x)Definition of a One-to-One FunctionA function f is one-to-one if, for a and b in its domain, f a f b impliesthat a b.yExistence of an Inverse FunctionA function f has an inverse function f 1 if and only if f is one-to-one.3From its graph, it is easy to tell whether a function of x is one-to-one. Simplycheck to see that every horizontal line intersects the graph of the function at mostonce. This is called the Horizontal Line Test. For instance, Figure 1.86 showsthe graph of y x2. On the graph, you can find a horizontal line that intersectsthe graph twice.Two special types of functions that pass the Horizontal Line Test are thosethat are increasing or decreasing on their entire domains.1. If f is increasing on its entire domain, then f is one-to-one.y x22( 1, 1) 21(1, 1)x 112 1Figure 1.86 f x ⴝ x 2 is notone-to-one.2. If f is decreasing on its entire domain, then f is one-to-one.Example 6 Testing for One-to-One FunctionsIs the function f x x 1 one-to-one?Algebraic SolutionGraphical SolutionLet a and b be nonnegative real numbers with f a f b .Use a graphing utility to graph the functiony x 1. From Figure 1.87, you can see that ahorizontal line will intersect the graph at most onceand the function is increasing. So, f is one-to-one anddoes have an inverse function. a 1 b 1Set f a f b . a ba bSo, f a f b implies that a b. You can conclude that f isone-to-one and does have an inverse function.5 2x 17 1Now try Exercise 55.y Figure 1.87

333371 0107.qxd12/27/06152Chapter 110:36 AMPage 152Functions and Their GraphsFinding Inverse Functions AlgebraicallyFor simple functions, you can find inverse functions by inspection. For morecomplicated functions, however, it is best to use the following guidelines.Finding an Inverse Function1. Use the Horizontal Line Test to decide whether f has an inversefunction.2. In the equation for f x , replace f x by y.3. Interchange the roles of x and y, and solve for y.4. Replace y by f 1 x in the new equation.5. Verify that f and f 1 are inverse functions of each other by showingthat the domain of f is equal to the range of f 1, the range of f isequal to the domain of f 1, and f f 1 x x and f 1 f x x.The function f with an implied domain of all real numbers may not pass theHorizontal Line Test. In this case, the domain of f may be restricted so that f doeshave an inverse function. For instance, if the domain of f x x2 is restricted tothe nonnegative real numbers, then f does have an inverse function.TECHNOLOGY TIPMany graphing utilities havea built-in feature for drawing aninverse function. To see how thisworks, consider the functionf x x . The inverse functionof f is given by f 1 x x2,x 0. Enter the functiony1 x. Then graph it in thestandard viewing window anduse the draw inverse feature.You should obtain the figurebelow, which shows both f andits inverse function f 1. Forinstructions on how to usethe draw inverse feature, seeAppendix A; for specifickeystrokes, go to this textbook’sOnline Study Center.f 1(x) x2, x 010Example 7 Finding an Inverse Function Algebraically 105 3x.Find the inverse function of f x 210 10Solutionf(x) xThe graph of f in Figure 1.88 passes the Horizontal Line Test. So you know thatf is one-to-one and has an inverse function.f x 5 3x25 3xy 2x 5 3y2Write original function.Replace f x by y.Interchange x and y.Multiply each side by 2.3y 5 2xIsolate the y-term.f 1 x 5 2x35 2x332x 5 3y5 2xy 3f 1(x) 24 1f(x) 5 3x2Figure 1.88Solve for y.Replace y by f 1 x .The domains and ranges of f and f 1 consist of all real numbers. Verify thatf f 1 x x and f 1 f x x.Now try Exercise 59.The draw inverse feature is particularlyuseful if you cannot find an expression forthe inverse function of a given function.For example, it would be very difficult todetermine the equation for the inversefunction of the one-to-one functionf x 14 x 5 14 x 3 12 x 1.However, it is easy to use the techniqueoutlined above to obtain the graph of theinverse function.