449 4.2 Exponential Functions Exponential Functions We now define a function ƒ1x2 = ax whose domain is the set of all real numbers. Notice how the independent variable x appears in the exponent in this function. Exponential Function If a 70 and a≠1, then the exponential function with base a is ƒ1x2 =ax. NOTE The restrictions a 70 and a≠1 in the definition of an exponential function are important. Consider the outcome of breaking each restriction. If a 60, say a = -2, and we let x = 1 2 , then ƒ A 1 2B = 1-221/2 = 2-2, which is not a real number. If a = 1, then the function becomes the constant function ƒ1x2 = 1x = 1, which is not an exponential function. EXAMPLE 1 Evaluating an Exponential Function For ƒ1x2 = 2x, find each of the following. Round to the nearest thousandth as needed. (a) ƒ1-12 (b) ƒ132 (c) ƒa 5 2b (d) ƒ14.922 SOLUTION (a) ƒ1-12 = 2-1 = 1 2 Replace x with -1. (b) ƒ132 = 23 = 8 (c) ƒ a 5 2b = 25/2 = 12521/2 = 321/2 = 232 = 216 # 2 = 422 (d) ƒ14.922 = 24.92 ≈30.274 Use a calculator. Round to the nearest thousandth. S Now Try Exercises 13, 19, and 23. We repeat the final graph of y = 2x (with real numbers as domain) from Figure 13 and summarize important details of the function ƒ1x2 = 2x here. • The y-intercept is 10, 12. • Because 2x 70 for all x and 2x S0 as x S-∞, the x-axis is a horizontal asymptote. • As the graph suggests, the domain of the function is 1-∞, ∞2 and the range is 10, ∞2. • The function is increasing on its entire domain. Therefore, it is one-to-one. These observations lead to the following generalizations about the graphs of exponential functions. y –2 2 f(x) = 2x 8 (2, 4) (1, 2) (0, 1) 6 4 2 x Q–1, R 1 2 Figure 13 (repeated) Graph of ƒ1x2 =2x with domain 1 −H, H2
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