384 CHAPTER 3 Polynomial and Rational Functions As x increases without bound, ƒ1x2 approaches 0, as suggested by the tables in Figure 48. Again, function values are all positive. The x-axis is the horizontal asymptote of the graph. The graph of ƒ1x2 = 1 x2 is shown in Figure 49. As x approaches ∞, y1 = 1 x2 approaches 0 through positive values. As x approaches -∞, y1 = 1 x2 approaches 0 through positive values. Figure 48 Rational Function ƒ1x2 = 1 x2 Domain: 1-∞, 02 ´10, ∞2 Range: 10, ∞2 • ƒ1x2 = 1 x2 increases on the open interval 1-∞, 02 and decreases on the open interval 10, ∞2. • It is discontinuous at x = 0. • The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote. • It is an even function, and its graph is symmetric with respect to the y-axis. −2.1 −6.6 6.1 6.6 f(x) = 1 x2 x y 0 1 1 f(x) = 1 x2 x y {3 1 9 {2 1 4 {1 1 { 1 2 4 { 1 4 16 0 undefined Figure 49 −2.1 −8.6 6.1 4.6 g(x) = − 1 1 (x + 2)2 Figure 51 EXAMPLE 3 Graphing a Rational Function Graph g1x2 = 1 1x + 222 - 1. Give the domain and range and the largest open intervals of the domain over which the function is increasing or decreasing. SOLUTION The function g1x2 = 1 1x + 222 - 1 is equivalent to g1x2 = ƒ1x + 22 - 1, where ƒ1x2 = 1 x2 . This indicates that the graph will be shifted to the left 2 units and down 1 unit. The horizontal shift affects the domain, now 1-∞, -22 ´1-2, ∞2. The vertical shift affects the range, now 1-1, ∞2. The vertical asymptote has equation x = -2, and the horizontal asymptote has equation y = -1. A traditional graph is shown in Figure 50, with a calculator graph in Figure 51. Both graphs show that this function is increasing on 1-∞, -22 and decreasing on 1-2, ∞2. S Now Try Exercise 27. –3 –1 1 x y x = –2 y = –1 4 g(x) = 1 (x + 2)2 0 – 1 This is the graph of y = shifted to the left 2 units and down 1 unit. 1 x2 Figure 50
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