288 CHAPTER 5 Exponential and Logarithmic Functions Procedure for Finding the Inverse of a One-to-One Function Step 1 In ( ) = y f x , interchange the variables x and y to obtain ( ) = x f 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 :1 ( ) = − y f x 1 Step 3 Check the result by showing that ( ) ( ) ( ) ( ) = = − − f f x x f f x x and 1 1 Step 3 Check the result by showing that f f x x 1( ) ( ) = − and f f x x. 1 ( ) ( ) = − We verified that f and −f 1 are inverses in Example 6(b). The graphs of ( ) = + f x x2 3 and its inverse f x x 1 2 3 1( ) ( ) = − − are shown in Figure 16. Note the symmetry of the graphs with respect to the line = y x. Figure 16 y 5 x f(x) 5 2x 1 3 x y 25 5 5 25 f 21(x) 5 (x 2 3) 1 – 2 The inverse function is ( ) = + − ≠ −f x x x x 1 2 2 1 Replace y by ( ) −f x . 1 Finding the Inverse of a Function Defined by an Equation The function ( ) = + − ≠ f x x x x 2 1 1 1 is one-to-one. Find its inverse function and check the result. Solution EXAMPLE 9 Step 1 Replace ( ) f x with y and interchange the variables x and y in = + − y x x 2 1 1 to obtain = + − x y y 2 1 1 Step 2 Solve for y . ( ) ( ) = + − − = + − = + − = + − = + = + − x y y x y y xy x y xy y x x y x y x x 2 1 1 1 2 1 2 1 2 1 2 1 1 2 Multiply both sides by −y 1. Use the Distributive Property. Subtract y2 from both sides; add x to both sides. Factor. Divide by −x 2.
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