300 CHAPTER 5 Exponential and Logarithmic Functions Figures 21 and 22 illustrate the graphs of two other exponential functions whose bases are larger than 1. Notice that the larger the base, the steeper the graph is when > x 0, and when < x 0, the larger the base, the closer the graph is to the x -axis. Graphing an Exponential Function Graph the exponential function: ( ) ( ) = f x 1 2 x EXAMPLE 4 Properties of the Exponential Function f x , 1 x ( ) = > a a • The domain is the set of all real numbers, or ( ) −∞ ∞, using interval notation; the range is the set of positive real numbers, or ( )∞ 0, using interval notation. • There are no x -intercepts; the y -intercept is 1. • The x -axis ( ) = y 0 is a horizontal asymptote of the graph of f as →−∞ x . • ( ) = > f x a a , 1, x is an increasing function and is one-to-one. • The graph of f contains the points ( ) ( ) − a 1, 1 , 0, 1 and ( )a 1, . • The graph of f is smooth and continuous, with no corners or gaps. See Figure 23. Figure 23 ( ) = > f x a a , 1 x y 5 0 x y (0, 1) 21, ) 1 a (1, a) ( Figure 21 y = 0 x y = 6 x y = 3 x y –3 3 (1, 3) (1, 6) (0, 1) 6 3 ( ) –1, 1 – 6 ( ) –1, 1 – 3 Figure 24 ( ) ( ) = f x 1 2 x x y –3 3 (–2, 4) (–1, 2) (0, 1) 1, 3, ( ) ( ) ( ) 2, 6 3 1 – 2 1 – 4 1 – 8 y 5 0 (a) Figure 22 10 −1 −4 Y1 = 3 x Y 2 = 6 x 4 (b) 1 0 −4 Y1 = 3 x Y2 = 6 x 0 Now consider ( ) = f x ax when < < a 0 1. Table 4 The domain of ( ) ( ) = f x 1 2 x is all real numbers. As before, locate some points on the graph as shown in Table 4 using Desmos. Because ( ) > 1 2 0 x for all x, the range of f is the interval ( )∞ 0, . The graph lies above the x -axis and has no x-intercepts. The y -intercept is 1. As ( ) ( ) →−∞ = x f x , 1 2 x grows very quickly. As →∞ x , the values of ( ) f x approach 0.The x -axis ( ) = y 0 is a horizontal asymptote of the graph of f as →∞ x . The function f is a decreasing function and so is one-to-one. Figure 24 shows the graph of f. Solution
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