435 4.1 Inverse Functions (b) We can determine that the function ƒ1x2 = 225 - x2 is not one-to-one by showing that different values of the domain correspond to the same value of the range. If we choose a = 3 and b = -3, then 3≠-3, but ƒ132 = 225 - 32 = 225 - 9 = 216 = 4 and ƒ1-32 = 225 - 1-322 = 225 - 9 = 4. Here, even though 3≠-3, we have ƒ132 = ƒ1-32 = 4. By the definition, ƒ is not a one-to-one function. S Now Try Exercises 17 and 19. –5 5 5 x y f(x) = Ë25 – x 2 0 (–3, 4) (3, 4) Figure 2 Horizontal Line Test A function is one-to-one if every horizontal line intersects the graph of the function at most once. EXAMPLE 2 Using the Horizontal LineTest Determine whether each graph is the graph of a one-to-one function. (a) x y1 x1 (x1, y1) (x2, y1) (x3, y1) x2 x3 0 y (b) x y y1 y3 y2 x3 x2 x1 0 SOLUTION (a) Each point where the horizontal line intersects the graph has the same value of y but a different value of x. Because more than one different value of x (here three) lead to the same value of y, the function is not one-to-one. (b) Every horizontal line will intersect the graph at exactly one point, so this function is one-to-one. S Now Try Exercises 11 and 13. As illustrated in Example 1(b), a way to show that a function is not one- to-one is to produce a pair of different domain elements that lead to the same function value. There is a useful graphical test for this, the horizontal line test. The function graphed in Example 2(b) decreases on its entire domain. In general, a function that is either increasing or decreasing on its entire domain, such as ƒ1x2 =x3 or g1x2 =−x, must be one- to-one. NOTE In Example 1(b), the graph of the function is a semicircle, as shown in Figure 2. Because there is at least one horizontal line that intersects the graph in more than one point, this function is not one-to-one.
RkJQdWJsaXNoZXIy NjM5ODQ=