451 4.2 Exponential Functions In Figure 16, the graphs of several typical exponential functions illustrate these facts. x y –2 –1 1 2 3 –3 0 3 4 5 f(x) = 10x f(x) = 3x f(x) = 2x 1 2 f(x) = ( ) x f(x) = ( ) x 1 3 f(x) = ( ) x 1 10 For a > 1, the function is increasing. f(x) = ax Domain: (–∞, ∞); Range: (0, ∞) For 0 < a < 1, the function is decreasing. The x-axis is a horizontal asymptote. Figure 16 In summary, the graph of a function of the form ƒ1x2 = ax has the following features. Characteristics of the Graph of f 1x2 =a x 1. The points A -1, 1 aB, 10, 12, and 11, a2 are on the graph. 2. If a 71, then ƒ is an increasing function. If 0 6a 61, then ƒ is a decreasing function. 3. The x-axis is a horizontal asymptote. 4. The domain is 1-∞, ∞2, and the range is 10, ∞2. EXAMPLE 2 Graphing an Exponential Function Graph ƒ1x2 = A1 5B x . Give the domain and range. SOLUTION The y-intercept is 10, 12, and the x-axis is a horizontal asymptote. Plot a few ordered pairs, and draw a smooth curve through them as shown in Figure 17. x ƒ1x2 -2 25 -1 5 0 1 1 1 5 2 1 25 –2 –1 1 2 5 10 15 20 25 x y 0 (–2, 25) (–1, 5) (0, 1) (1, ) 1 5 f(x) = ( ) x 1 5 Figure 17 This function has domain 1-∞, ∞2, range 10, ∞2, and is one-to-one. It is decreasing on its entire domain. S Now Try Exercise 29.
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