450 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Recall that the graph of y = ƒ1-x2 is the graph of y = ƒ1x2 reflected across the y-axis. Thus, we have the following. If ƒ1x2 = 2x, then ƒ1-x2 = 2-x = 2-1# x = 12-12x = a 1 2b x . This is supported by the graphs in Figures 14 and 15. The graph of ƒ1x2 = 2x is typical of graphs of ƒ1x2 = ax where a 71. For larger values of a, the graphs rise more steeply, but the general shape is similar to the graph in Figure 14. When 0 6a 61, the graph decreases in a manner similar to the graph of ƒ1x2 = A1 2B x in Figure 15. Exponential Function f 1x2 =ax Domain: 1-∞, ∞2 Range: 10, ∞2 For ƒ1x2 = 2x: x ƒ1x2 -2 1 4 -1 1 2 0 1 1 2 2 4 3 8 • ƒ1x2 =ax, for a +1, is increasing and continuous on its entire domain, 1-∞, ∞2. • The x-axis is a horizontal asymptote as x S-∞. • The graph passes through the points A -1, 1 aB, 10, 12, and 11, a2. For ƒ1x2 = A1 2B x : x ƒ1x2 -3 8 -2 4 -1 2 0 1 1 1 2 2 1 4 • ƒ1x2 =ax, for 0 *a *1, is decreasing and continuous on its entire domain, 1-∞, ∞2. • The x-axis is a horizontal asymptote as x S∞. • The graph passes through the points A -1, 1 aB, 10, 12, and 11, a2. This is the general behavior seen on a calculator graph for any base a, for a +1. f(x) = ax, a > 1 x y 0 (0, 1) (1, a) (–1, ) a 1 f(x) = ax, a > 1 Figure 14 Figure 15 f(x) = ax, 0 < a < 1 x y 0 (0, 1) (1, a) (–1, ) a 1 f(x) = ax, 0 < a < 1 This is the general behavior seen on a calculator graph for any base a, for 0 *a *1.
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