SECTION 5.2 One-to-One Functions; Inverse Functions 287 Now Work PROBLEMS 41 AND 45 Verifying Inverse Functions Verify that the inverse of ( ) = − f x x 1 1 is ( ) = + −f x x 1 1. 1 For what values of x is ( ) ( ) = −f f x x? 1 For what values of x is ( ) ( ) = − f f x x? 1 Solution EXAMPLE 7 The domain of f is { } ≠ x x 1 and the domain of −f 1 is { } ≠ x x 0 . Now f f x f x x x x x 1 1 1 1 1 1 1 1 provided 1 1 1( ) ( ) ( ) = − = − + = − + = ≠ − − ( ) ( ) ( ) ( ) = + = + − = = ≠ − f f x f x x x x x 1 1 1 1 1 1 1 1 provided 0 1 5 Find the Inverse of a Function Defined by an Equation The fact that the graphs of a one-to-one function f and its inverse function −f 1 are symmetric with respect to the line = y x tells us more. It says that we can obtain −f 1 by interchanging the roles of x and y in f. If f is defined by the equation ( ) = y f x then −f 1 is defined by the equation ( ) = x f y The equation ( ) = x f y defines −f 1 implicitly. If we can solve this equation for y, we will have the explicit form of −f ,1 that is, ( ) = − y f x 1 Let’s use this procedure to find the inverse of ( ) = + f x x2 3. Because f is a linear function and is increasing, f is one-to-one and so has an inverse function. Need to Review? Implicit functions are discussed in Section 2.1, p. 68. (continued) How to Find the Inverse of a Function Defined by an Equation Find the inverse of ( ) = + f x x2 3. Graph f and −f 1 on the same coordinate axes. EXAMPLE 8 Replace ( ) f x with y in ( ) = + f x x2 3 and obtain = + y x2 3. Now interchange the variables x and y to obtain x y2 3 = + This equation defines the inverse function −f 1 implicitly. Step-by-Step Solution Step 1 Replace f x( ) with y. Then interchange the variables x and y. This equation defines the inverse function f 1− implicitly. Step 2 If possible, solve the implicit equation for y in terms of x to obtain the explicit form of f y f x , . 1 1( ) = − − To find the explicit form of the inverse, solve = + x y2 3 for y. x y y x y x y x 2 3 2 3 2 3 1 2 3 ( ) = + + = = − = − The explicit form of the inverse function −f 1 is f x x 1 2 3 1( ) ( ) = − − Subtract 3 from both sides. Multiply both sides by 1 2 .
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