284 CHAPTER 5 Exponential and Logarithmic Functions We begin with finding inverses of functions defined by mappings. Finding the Inverse of a Function Defined by a Mapping Find the inverse of the function defined by the map. Let the domain of the function represent certain states, and let the range represent the states’ populations (in millions). Find the domain and the range of the inverse function. EXAMPLE 3 Indiana Washington South Dakota North Carolina Oklahoma State 6.9 7.8 0.9 11.0 4.1 Population (in millions) Indiana Washington South Dakota North Carolina Oklahoma State 6.9 7.8 0.9 11.0 4.1 Population (in millions) Solution The function is one-to-one. To find the inverse function, interchange the elements in the domain with the elements in the range. For example, the function receives as input Indiana and outputs 6.9 million. So the inverse receives as input 6.9 million and outputs Indiana. The inverse function is shown next. The domain of the inverse function is { } 6.9, 7.8, 0.9, 11.0, 4.1 .The range of the inverse function is { } Indiana, Washington, South Dakota, North Carolina, Oklahoma . If the function f is a set of ordered pairs ( ) x y , , then the inverse function of f , denoted −f ,1 is the set of ordered pairs ( ) y x , . Finding the Inverse of a Function Defined by a Set of Ordered Pairs Find the inverse of the following one-to-one function: ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } − − − − − − 3, 27, 2, 8, 1, 1, 0,0, 1,1, 2,8, 3,27 State the domain and the range of the function and its inverse. EXAMPLE 4 Solution The inverse of the given function is found by interchanging the entries in each ordered pair and so is given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } − − − − − − 27, 3, 8, 2, 1, 1, 0,0, 1,1, 8,2, 27,3 The domain of the function is { } − − − 3, 2, 1, 0, 1, 2, 3 . The range of the function is { } − − − 27, 8, 1, 0, 1, 8, 27 . The domain of the inverse function is { } − − − 27, 8, 1, 0, 1, 8, 27 . The range of the inverse function is { } − − − 3, 2, 1, 0, 1, 2, 3 . Now Work PROBLEMS 27 AND 31
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