SECTION 5.2 One-to-One Functions; Inverse Functions 283 2 Determine the Inverse of a Function Defined by a Mapping or a Set of Ordered Pairs One-to-one functions ( ) = y f x have an important property. Corresponding to each x in the domain of f, there is exactly one y in the range because f is a function. And corresponding to each y in the range of f, there is exactly one x in the domain because f is one-to-one.The function represented by the correspondence from the range back to the domain is called the inverse function of f. Using the Horizontal-line Test For each function, use its graph to determine whether the function is one-to-one. (a) ( ) = f x x2 (b) ( ) = g x x3 EXAMPLE 2 Solution (a) Figure 10(a) illustrates the horizontal-line test for ( ) = f x x .2 The horizontal line = y 1 intersects the graph of f twice, at ( ) 1, 1 and at ( ) −1, 1 , so f is not one-to-one. (b) Figure 10(b) illustrates the horizontal-line test for ( ) = g x x .3 Because every horizontal line intersects the graph of g exactly once, the function g is one-to-one. THEOREM • A function that is increasing on an interval I is a one-to-one function on I . • A function that is decreasing on an interval I is a one-to-one function on I . Figure 10 Every horizontal line intersects the graph exactly once; g is one-to-one. A horizontal line intersects the graph twice; f is not one-to-one. x y 23 3 (1, 1) y 5 x 2 (21, 1) 3 23 (a) (b) x y 23 3 y 5 x 3 3 23 y 5 1 Now Work PROBLEM 21 Look more closely at the one-to-one function ( ) = g x x3 in Example 2(b). This function is an increasing function. Because an increasing (or decreasing) function always has different y -values for unequal x -values, it follows that a function that is increasing (or decreasing) over its domain is also a one-to-one function. DEFINITION Inverse Function Suppose ( ) = y f x is a one-to-one function. The correspondence from the range of f to the domain of f is called the inverse function of f . The symbol −f 1 is used to denote the inverse function of f. In other words, if ( ) = y f x is a one-to-one function, then f has an inverse function −f 1 and ( ) = − x f y . 1 In Words Suppose that f is a one-to-one function so that the input 5 corresponds to the output 10. For the inverse function f ,1− the input 10 will correspond to the output 5.
RkJQdWJsaXNoZXIy NjM5ODQ=