6.10 Functions and Their Graphs 397 All exponential functions of the form = = > y a f x a a , or ( ) where 1, x x will have the general shape of the graph illustrated in Fig. 6.49. y x y 5 2 x 4 6 5 3 8 9 7 2 22 1 2 3 4 5 21 23 21 (2, 4) (1, 2) (0, 1) (21, )1– 2 (3, 8) 1 Figure 6.49 x x x x x x x 3, 2, 1, 0, 1, 2, 3, = − = − = − = = = = y y y y y y y y 2 2 1 2 1 8 2 1 2 1 4 2 1 2 1 2 2 1 2 2 2 4 2 8 x 3 3 2 2 1 1 0 1 2 3 = = = = = = = = = = = = = = = = = = − − − Now try Exercise 61 x y −3 1 8 −2 1 4 −1 1 2 0 1 1 2 2 4 3 8 The domain is all real numbers, .R The range is > y 0. Note that y can never have a value of 0. 7 y 4 6 5 3 8 9 7 2 22 1 2 3 21 23 24 x 21 1 2 y 5 x (23, 8) (22, 4) (21, 2) (0, 1) (1, )1– 2 1 Figure 6.50 Example 11 Graphing an Exponential Function with a Base Between 0 and 1 a) Graph ( ) = y . x 1 2 b) Determine the domain and range of the function. Solution a) We begin by substituting values for x and calculating values for y. We then plot the ordered pairs and use these points to graph the function. To evaluate a fraction with a negative exponent, we use the fact that ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝⎜ ⎞ ⎠⎟ − a b b a x x For example, ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ = − 1 2 2 1 8 3 3 Then x x x x x x x 3, 2, 1, 0, 1, 2, 3, = − = − = − = = = = = ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ = = = ⎛ ⎝ ⎞ ⎠ = = = ⎛ ⎝ ⎞ ⎠ = = = ⎛ ⎝ ⎞ ⎠ = = ⎛ ⎝ ⎞ ⎠ = = ⎛ ⎝ ⎞ ⎠ = = ⎛ ⎝ ⎞ ⎠ = − − − y y y y y y y y 1 2 1 2 2 8 1 2 2 4 1 2 2 2 1 2 1 1 2 1 2 1 2 1 4 1 2 1 8 x 3 3 2 2 1 1 0 1 2 3 Now try Exercise 63 x y −3 8 −2 4 −1 2 0 1 1 1 2 2 1 4 3 1 8 The graph is illustrated in Fig. 6.50. b) The domain is the set of all real numbers, .R The range is > y 0. 7
RkJQdWJsaXNoZXIy NjM5ODQ=