pg180 [V] G2 5-36058 / HCG / Cannon & Elich clb 11-22-95 MP1 180 Chapter 3 Polynomial and Rational Functions 3.4 RATIONAL FUNCTIONS What is mathematics about? I think it’s really summed up in what I frequently tell my classes. That is that proofs really aren’t there to convince you that something is true—they’re there to show you why it is true. That’s what it’s all about—it’s to try to figure out how it’s all tied together. Freeman Dyson In this section we consider rational functions, which are defined as quotients of polynomial functions. Definition: rational function Suppose p and q are polynomial functions, where q is not the zero function. Then the function f given by The only idea of real p~x! mathematics that I had f ~x! 5 came from Men of q~x! Mathematics. In it, I got my p first glimpse of a mathe- is called a rational function, sometimes written f 5 q . The domain of f matician per se. I cannot consists of all real numbers for which q~x! 5/ 0. overemphasize the p If p~x! and q~x! have no common factors, then we say that q is in importance of such books reduced form. about mathematics in the intellectual life of a student like myself completely For our discussion in this section, we assume that all rational functions are reduced out of contact with unless we specify to the contrary. The most significant information for analyzing research mathematicians. the behavior of a rational function is the set of zeros of the polynomial functions Julia Robinson in the numerator and the denominator. Togetafeelingforthemeaningofzerosinthedenominator,welookata variety of graphs. In Example 1 we consider pairs of graphs, each consisting of a polynomial and its reciprocal. cEXAMPLE 1 Functions and reciprocals Sketch the graphs of both func- tions, together with the horizontal lines y 5 1andy521, in the specified win- dow. Then describe where the graphs of f and g meet, and where the graph of g goes off-scale (out of sight in the window). 1 (a) f ~x! 5 x 2 2 2x, g~x! 5 . Decimal window x 2 2 2x y=x2–2x 1 y =1 (b) f ~x! 5 x 3 2 3x 2, g~x! 5 . Change y-range to @25, 5# x 3 2 3x 2 Solution (a) The graph of f is the solid parabola in Figure 23, and y 5 g~x! is the dotted curve. It appears that the graphs of f and g meet where f~x! 561. The graph y=–1 of g appearstogooff-scaleatx50andx52; that is, where f ~x! 5 0. (b) In Figure 24a, again the graphs of f and g meet where f~x! 561. The graph of g appearstogooff-scaleatx50. It is less clear what happens to g near [– 5, 5] by [– 3.5, 3.5] x 5 3(theotherpointwheref~x!50),sowelookcloserbyreducingthe FIGURE 23 x-range to @2, 4# (Figure 24b). Now we still see the intersections where pg181 [R] G1 5-36058 / HCG / Cannon & Elich jb 11-13-95 QC2 3.4 Rational Functions 181 3 2 y=x –3x y = x3 – 3x2 y =1 y = 1 y=–1 y = – 1 [– 5, 5] by [– 5, 5] [2, 4] by [– 5, 5] (a) (b) FIGURE 24 f ~x! 561, but the graph of g clearly goes off-scale as x nears3.Yourcalculator mayshowanearlyverticalcolumnofpixelsnear3,thisisa“false (calculator) asymptote.” It occurs because the calculator connects widely separated y-values in adjacent columns, but it is not part of the graph of g. b TECHNOLOGY TIP r Rational functions and parentheses In graphing rational functions on a calculator, one of the most common errors involves the use of parentheses. We are so accustomed to using the fraction bar as a separator that we can get careless with our calculator. In 1 1 Example 1(a), g~x! 5 5 . On the calculator we could enter x 2 2 2x x~x 1 2! Y 5 1y~X2 2 2X) or Y 5 1y~X~X 2 2)) or Y 5 1y~X*~X 2 2)) but NOT Y 5 1yX2 2 2X or Y 5 1yX*~X 2 2). Every calculator is programmed to handle operations differently. Try each of the above on your calculator and make sure you know how to get a graph that looks like Figure 23. Vertical Asymptotes The kind of “off-scale” behavior we observed in the calculator graphs in Example 1 deserves closer examination. When we plot y 5 1yx in a decimal window (Fig- y = 1 ure 25), the graph goes off-scale in both directions, with the graph seeming to get x closer and closer to the y-axis. If, however, we increase the y -range to @215, 15#, the calculator graph climbs up the y-axis and then stops; the highest point is ( 1 ,10). 10 1 Doesthegraphreallystopat(10, 10), or does it keep climbing? We could see more by zooming in near (0, 10), but the calculator is not the best tool for answering the question. Looking at the equation y 5 1yx,itiseasytoseethatwecangeta y-value of 100 (at x 5 1y100), or 1000, or ten million, by taking x-values small [– 5, 5] by [– 3.5, 3.5] enough. There is no highest point. Taking positive x-valuescloserandcloserto0, the y-values keep growing without bound. FIGURE 25 We use arrows to describe such behavior and write: as x A 01, y A ` (or 1 1yx A `), or more compactly, lim x 5`. xA01 pg182 [V] G2 5-36058 / HCG / Cannon & Elich jb 11-13-95 QC2 182 Chapter 3 Polynomial and Rational Functions Arrow notation x A a1 means that x approaches a from above; that is, x takesonvalues near a, but greater than a ~such as a 1 0.01, a 1 0.001,...). xAa2 means that x approaches a from below; that is, x takes on values near a, but less than a (such as a 2 0.01, a 2 0.001,...). Similarly, x A ` or x A 2` means that x assumes larger and larger positive or negative values, respectively. The same notation is used to indicate functional behavior. In calculus the concept of limit has a very important, precise meaning. Here we use the notation only for the intuitive notion embodied in our arrows. Looking back at Figure 23, we see the following: lim g~x! 5`, lim g~x! 52`, xA02 xA01 lim g~x! 52`, lim g~x! 5`. xA22 xA21 The vertical lines x 5 0(they-axis) and x 5 2 are called vertical asymptotes for the curve y 5 g~x!. Without attempting a more precise definition, we say that a line is an asymptote foracurveifthe distance between the curve and the line goes to zero as we move out along the line. From the graphs in Example 1, it is clear that each reciprocal function 1yf~x! has a vertical asymptote at each zero of f ~x!, and furthermore, that the x-axis is a horizontal asymptote for each reciprocal function since g~x! A 0asxA`and g~x! A 0asxA2`. Asymptotes for rational functions can be vertical, horizontal, or oblique lines, as illustrated in Figure 26. y y Vertical Horizontal asymptote asymptote x x Vertical Oblique asymptote asymptote FIGURE 26 Shifts and reflections are also useful in graphing rational functions. 1 cEXAMPLE 2 Shifts and reflections Usethegraphoff~x!5x to graph 1 2 1 1 x (a) y 5 (b) y 5 (c) y 5 Strategy: Try to relate each x 2 1 2 2 x x 1 function to f~x! 5 x .In(a) Solution f ~x 2 1! gives a horizontal 1 1 (b) (a) Since f~x 2 1! 5 x 2 1 ,graphy5x21 by translating the graph of f one unit translation. In factor out 1 22toget22·f~x22!.In to the right. As a useful check, observe that y 5 x 2 1 has a vertical asymptote 1 (c) y 5 x 1 1, for a vertical where the denominator is 0, at x 5 1.Theresultofthetranslationisshownin translation. Figure 27a. pg183 [R] G1 5-36058 / HCG / Cannon & Elich jb 11-13-95 QC2 3.4 Rational Functions 183 y y y 1+x 4 4 4 y = x 1 3 3 3 y = y = 2 x=2 y=1 x –1 2 2–x 2 2 1 1 –4–3–2 –11 –4–3 –2–1 1 2 3 4 x x x 1 2 3 4 1 2 3 4 –4 –3–2 –1 –1 –1 –1 –2 –2 –2 x=1 –3 –3 –3 –4 –4 –4 (a) (b) (c) FIGURE 27 1 Shifts and reflections of y 5 . x 2 22 (b) If we factor out 21fromthedenominator,then22x5x22,so 2 22x522f~x22!.Translatethegraphofftwo units to the right, reflect it through the x-axisandstretchitverticallybyafactorof2.Plottingafewpoints gives the graph shown in Figure 27b. 1 1 x 1 1 x (c) To relate to f ~x!, rewrite as x x 1 1 x 1 x 1 5 1 5 1 1 5 f~x! 1 1. x x x x Translate the graph of f one unit up, as shown in Figure 27c. b Graphing Other Rational Functions All of the rational functions we have graphed thus far are either reciprocals or shifts of reciprocals of polynomial functions, where we have observed that there is a vertical asymptote at every zero of the denominator. What is the significance of p(x! the zeros of the numerator? Suppose f~x! 5 and f is in reduced form (so that q~x! p and q have no common zeros).