382 CHAPTER 3 Polynomial and Rational Functions The graph of ƒ1x2 = 1 x is shown in Figure 43. Reciprocal Function ƒ1x2 = 1 x Domain: 1-∞, 02 ´10, ∞2 Range: 1-∞, 02 ´10, ∞2 • ƒ1x2 = 1 x decreases on the open intervals 1-∞, 02 and 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 odd function, and its graph is symmetric with respect to the origin. x y -2 - 1 2 -1 -1 - 1 2 -2 0 undefined 1 2 2 1 1 2 1 2 Figure 43 –2 1 –2 1 x y 0 f(x) = 1 x Vertical asymptote Horizontal asymptote −4.1 −6.6 4.1 6.6 f(x) = 1 x The graph of y = 1 x can be translated and/or reflected. EXAMPLE 1 Graphing a Rational Function Graph y = - 2 x . Give the domain and range and the largest open intervals of the domain over which the function is increasing or decreasing. SOLUTION The expression - 2 x can be written as -2 A 1 xB or 2 A 1 -xB, indicating that the graph may be obtained by stretching the graph of y = 1 x vertically by a factor of 2 and reflecting it across either the x-axis or the y-axis. The x- and y-axes remain the horizontal and vertical asymptotes. The domain and range are both still 1-∞, 02 ´10, ∞2. See Figure 44. −4.1 −6.6 4.1 6.6 y = − 2 x The graph in Figure 44 is shown here using a decimal window. Using a nondecimal window may produce an extraneous vertical line that is not part of the graph. –2 2 –2 2 x y 0 y = – 2x Figure 44 The graph shows that ƒ1x2 is increasing on both sides of its vertical asymptote. Thus, it is increasing on 1-∞, 02 and 10, ∞2. S Now Try Exercise 17.
RkJQdWJsaXNoZXIy NjM5ODQ=