383 3.5 Rational Functions: Graphs, Applications, and Models y 1 2 –2 x x = –1 0 f(x) = 2 x + 1 Figure 45 −4.1 −6.6 4.1 6.6 f(x) = 2 x + 1 Figure 46 ALGEBRAIC SOLUTION The expression 2 x + 1 can be written as 2Q 1 x + 1R , indicating that the graph may be obtained by shifting the graph of y = 1 x to the left 1 unit and stretching it vertically by a factor of 2. See Figure 45. The horizontal shift affects the domain, which is now 1-∞, -12 ´1-1, ∞2. The line x = -1 is the vertical asymptote, and the line y = 0 (the x-axis) remains the horizontal asymptote. The range is still 1-∞, 02 ´10, ∞2. The graph shows that ƒ1x2 is decreasing on both sides of its vertical asymptote. Thus, it is decreasing on 1-∞, -12 and 1-1, ∞2. GRAPHING CALCULATOR SOLUTION When entering this rational function into the function editor of a calculator, make sure that the numerator is 2 and the denominator is the entire expression 1x +12. The graph of this function has a vertical asymptote at x = -1 and a horizontal asymptote at y = 0, so it is reasonable to choose a viewing window that contains the locations of both asymptotes as well as enough of the graph to determine its basic characteristics. See Figure 46. EXAMPLE 2 Graphing a Rational Function Graph ƒ1x2 = 2 x + 1 . Give the domain and range and the largest open intervals of the domain over which the function is increasing or decreasing. S Now Try Exercise 19. The Function f 1x2 = 1 x2 The rational function ƒ1x2 = 1 x2 Rational function also has domain 1-∞, 02 ´10, ∞2. We can use the table feature of a graphing calculator to examine values of ƒ1x2 for some x-values close to 0. See Figure 47. As x approaches 0 from the left, y1 = 1 x2 approaches ∞. As x approaches 0 from the right, y1 = 1 x2 approaches ∞. Figure 47 The tables suggest that ƒ1x2 increases without bound as x gets closer and closer to 0. Notice that as x approaches 0 from either side, function values are all positive and there is symmetry with respect to the y-axis. Thus, ƒ1x2 S∞ as x S0. The y-axis 1x = 02 is the vertical asymptote.
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