302 CHAPTER 5 Exponential and Logarithmic Functions Graphing an Exponential Function Using Transformations Graph ( ) = − − f x 2 3 x and determine the domain, range, horizontal asymptote, and y -intercept of f. Solution EXAMPLE 5 Begin with the graph of = y 2 .x Figure 29 shows the steps. As Figure 29(c) illustrates, the domain of ( ) = − − f x 2 3 x is the interval ( ) −∞ ∞, and the range is the interval ( ) − ∞ 3, . The horizontal asymptote of the graph of f is the line = − y 3. The y -intercept is ( ) = − = − = − f 0 2 3 1 3 2. 0 Check: Graph = − − Y 2 3 x 1 to verify the graph obtained in Figure 29(c). Figure 29 (3, 8) (2, 4) (1, 2) (0, 1) ( ) ( ) ( ) 21, 1 – 2 x y 3 10 (23, 8) (22, 4) (21, 2) (0, 1) 1, 1 – 2 x y 23 1 10 (23, 5) (22, 1) (0, 22) (21, 21) 1, 22 5 2 x y 2 10 24 (a) y 5 2 x (b) y 5 22x (c) y 5 22x 2 3 y 5 23 y 5 0 y 5 0 Replace x by 2x; reflect about y-axis Subtract 3; shift down 3 units Now Work PROBLEM 47 3 Define the Number e Many problems that occur in nature require the use of an exponential function whose base is a certain irrational number, symbolized by the letter e. One way of arriving at this important number e is given next. Historical Feature T he number e is called Euler's number in honor of the Swiss mathematician Leonard Euler (1707—1783). DEFINITION Number e The number e is defined as the number that the expression ( ) + n 1 1 n (2) approaches as →∞ n . In calculus, this is expressed, using limit notation, as ( ) = + →∞ e n lim 1 1 n n
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