SECTION 5.2 One-to-One Functions; Inverse Functions 289 Step 3 Check: Exploration In Example 9, we found that if ( ) = + − f x x x 2 1 1 , then ( ) = + − −f x x x 1 2 . 1 Compare the vertical and horizontal asymptotes of f and −f .1 Result The vertical asymptote of f is = x 1, and the horizontal asymptote is = y 2. The vertical asymptote of −f 1 is = x 2, and the horizontal asymptote is = y 1. Finding the Inverse of a Domain-restricted Function Find the inverse of ( ) = = y f x x2 if ≥ x 0. Graph f and −f .1 Solution EXAMPLE 10 The function = y x2 is not one-to-one. [Refer to Example 2(a).] However, restricting the domain of this function to ≥ x 0, as indicated, results in a new function that is increasing and therefore is one-to-one. Consequently, the function defined by ( ) = = ≥ y f x x x , 0, 2 has an inverse function, −f .1 Follow the steps to find −f .1 Step 1 In the equation = ≥ y x x , 0, 2 interchange the variables x and y. The result is = ≥ x y y 0 2 This equation defines the inverse function implicitly. Step 2 Solve for y to get the explicit form of the inverse. Because ≥ y 0, only one solution for y is obtained: = y x. So ( ) = −f x x. 1 Step 3 Check: f f x f x x x x x f f x f x x x because 0 1 1 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = ≥ = = = − − − Figure 17 illustrates the graphs of ( ) = ≥ f x x x , 0, 2 and ( ) = −f x x. 1 Figure 17 x y 2 2 f 21(x) 5 x y 5 x f (x) 5 x2, x $ 0 SUMMARY • If a function f is one-to-one, then it has an inverse function −f .1 • Domain of = f Range of −f ;1 Range of = f Domain of −f .1 • To verify that −f 1 is the inverse of f, show that ( ) ( ) = −f f x x 1 for every x in the domain of f and that ( ) ( ) = − f f x x 1 for every x in the domain of −f .1 • The graphs of f and −f 1 are symmetric with respect to the line = y x. f f x f x x x x x x x x x x x x x 2 1 1 2 1 1 1 2 1 1 2 2 1 1 2 1 2 1 3 3 , 1 1 1( ) ( ) ( ) ( ) = + − = + − + + − − = + + − + − − = = ≠ − − f f x f x x x x x x x x x x x x x 1 2 2 1 2 1 1 2 1 2 1 2 1 2 3 3 , 2 1 ( ) ( ) ( ) ( ) ( ) = + − = ⋅ + − + + − − = + + − + − − = = ≠ − Now Work PROBLEMS 53 AND 67 If a function is not one-to-one, it has no inverse function. Sometimes, though, an appropriate restriction on the domain of such a function yields a new function that is one-to-one. Then the function defined on the restricted domain has an inverse function.
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