282 CHAPTER 5 Exponential and Logarithmic Functions Put another way, a function f is one-to-one if no y in the range is the image of more than one x in the domain. A function is not one-to-one if any two (or more) different elements in the domain correspond to the same element in the range. So the function in Figure 7 is not one-to-one because two different elements in the domain, dog and cat , both correspond to 11 (and also because three different elements in the domain correspond to 10). Figure 8 illustrates the distinction among one-to-one functions, functions that are not one-to-one, and relations that are not functions. Determining Whether a Function Is One-to-One Determine whether the following functions are one-to-one. (a) For the following function, the domain represents the ages of five males, and the range represents their HDL (good) cholesterol scores (mg/dL). EXAMPLE 1 Solution (a) The function is not one-to-one because there are two different inputs, 55 and 61, that correspond to the same output, 38. (b) The function is one-to-one because every distinct input corresponds to a different output. 38 42 46 55 61 Age 57 54 34 38 HDL Cholesterol Figure 8 x3 x1 y1 x2 y2 One-to-one function: Each x in the domain has one and only one image in the range. (a) y3 Domain Range x3 x1 y1 x2 Not a one-to-one function: y1 is the image of both x1 and x2. (b) y3 Domain Range x3 x1 y1 y2 Not a function: x1 has two images, y1 and y2. (c) y3 For functions defined by an equation ( ) = y f x and for which the graph of f is known, there is a simple test, called the horizontal-line test , to determine whether f is one-to-one. (b) ( ) ( ) ( ) ( ) ( ) { } − − 2, 6 , 1, 3 , 0, 2 , 1, 5 , 2, 8 Now Work PROBLEMS 13 AND 17 THEOREM Horizontal-line Test If every horizontal line intersects the graph of a function f in at most one point, then f is one-to-one. The reason why this test works can be seen in Figure 9, where the horizontal line = y h intersects the graph at two distinct points, ( ) x h , 1 and ( ) x h , . 2 Since h is the image of both x1 and ≠ x x x and , 2 1 2 f is not one-to-one. Based on Figure 9, we can state the horizontal-line test in another way: If the graph of any horizontal line intersects the graph of a function f at more than one point, then f is not one-to-one. Figure 9 ( ) ( ) = = f x f x h 1 2 and ≠ x x ; 1 2 f is not a one-to-one function. x h y (x1, h) (x2, h) y 5 f(x) y 5 h x1 x2
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