3.4 RATIONAL FUNCTIONS What Is Mathematics About? I Think It’S Really Summed up in What I Frequently Tell My Classes

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 p͑x͒ mathematics that I had f ͑x͒ ϭ came from Men of q͑x͒ Mathematics. In it, I got my p first glimpse of a mathe- is called a rational function, sometimes written f ϭ q . The domain of f matician per se. I cannot consists of all real numbers for which q͑x͒ ϭր 0. overemphasize the p If p͑x͒ and q͑x͒ 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.

᭤EXAMPLE 1 Functions and reciprocals Sketch the graphs of both functions, together with the horizontal lines y ϭ 1andyϭϪ1, in the speciﬁed window. 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 ͑x͒ ϭ x 2 Ϫ 2x, g͑x͒ ϭ . Decimal window x 2 Ϫ 2x y=x2Ð2x 1 y =1 (b) f ͑x͒ ϭ x 3 Ϫ 3x 2, g͑x͒ ϭ . Change y-range to ͓Ϫ5, 5͔ x 3 Ϫ 3x 2 Solution (a) The graph of f is the solid parabola in Figure 23, and y ϭ g͑x͒ is the dotted curve. It appears that the graphs of f and g meet where f͑x͒ ϭϮ1. The graph y=Ð1 of g appearstogooff-scaleatxϭ0andxϭ2; that is, where f ͑x͒ ϭ 0. (b) In Figure 24a, again the graphs of f and g meet where f͑x͒ ϭϮ1. The graph of g appearstogooff-scaleatxϭ0. It is less clear what happens to g near [Ð 5, 5] by [Ð 3.5, 3.5] x ϭ 3(theotherpointwheref͑x͒ϭ0),sowelookcloserbyreducingthe FIGURE 23 x-range to ͓2, 4͔ (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 Ð3x y = x3 Ð 3x2 y =1 y = 1

y=Ð1 y = Ð 1

[Ð 5, 5] by [Ð 5, 5] [2, 4] by [Ð 5, 5] (a) (b)

FIGURE 24

f ͑x͒ ϭϮ1, 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. ᭣

TECHNOLOGY TIP ᭜ 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), g͑x͒ ϭ ϭ . On the calculator we could enter x 2 Ϫ 2x x͑x ϩ 2͒ Y ϭ 1͑͞X2 Ϫ 2X) or Y ϭ 1͑͞X͑X Ϫ 2)) or Y ϭ 1͑͞X*͑X Ϫ 2)) but NOT Y ϭ 1͞X2 Ϫ 2X or Y ϭ 1͞X*͑X Ϫ 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 ϭ 1͞x 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 ͓Ϫ15, 15͔, 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 ϭ 1͞x,itiseasytoseethatwecangeta y-value of 100 (at x ϭ 1͞100), or 1000, or ten million, by taking x-values small [Ð 5, 5] by [Ð 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 0ϩ, y A ϱ (or 1 1͞x A ϱ), or more compactly, lim x ϭϱ. xA0ϩ pg182 [V] G2 5-36058 / HCG / Cannon & Elich jb 11-13-95 QC2

182 Chapter 3 Polynomial and Rational Functions

Arrow notation x A aϩ means that x approaches a from above; that is, x takesonvalues near a, but greater than a ͑such as a ϩ 0.01, a ϩ 0.001,...). xAaϪ means that x approaches a from below; that is, x takes on values near a, but less than a (such as a Ϫ 0.01, a Ϫ 0.001,...). Similarly, x A ϱ or x A Ϫϱ 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 g͑x͒ ϭϱ, lim g͑x͒ ϭϪϱ, xA0Ϫ xA0ϩ lim g͑x͒ ϭϪϱ, lim g͑x͒ ϭϱ. xA2Ϫ xA2ϩ The vertical lines x ϭ 0(they-axis) and x ϭ 2 are called vertical asymptotes for the curve y ϭ g͑x͒. 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 1͞f͑x͒ has a vertical asymptote at each zero of f ͑x͒, and furthermore, that the x-axis is a horizontal asymptote for each reciprocal function since g͑x͒ A 0asxAϱand g͑x͒ A 0asxAϪϱ. 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 ᭤EXAMPLE 2 Shifts and reﬂections Usethegraphoff͑x͒ϭx to graph 1 2 1 ϩ x (a) y ϭ (b) y ϭ (c) y ϭ Strategy: Try to relate each x Ϫ 1 2 Ϫ x x 1 function to f͑x͒ ϭ x .In(a) Solution f ͑x Ϫ 1͒ gives a horizontal 1 1 (b) (a) Since f͑x Ϫ 1͒ ϭ x Ϫ 1 ,graphyϭxϪ1 by translating the graph of f one unit translation. In factor out 1 Ϫ2togetϪ2·f͑xϪ2͒.In to the right. As a useful check, observe that y ϭ x Ϫ 1 has a vertical asymptote 1 (c) y ϭ x ϩ 1, for a vertical where the denominator is 0, at x ϭ 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 Ð1 2 2Ðx 2 2 1 1 Ð4Ð3Ð2 Ð11 Ð4Ð3 Ð2Ð1 1 2 3 4 x x x 1 2 3 4 1 2 3 4 Ð4 Ð3Ð2 Ð1 Ð1 Ð1 Ð1 Ð2 Ð2 Ð2 x=1 Ð3 Ð3 Ð3 Ð4 Ð4 Ð4

(a) (b) (c)

FIGURE 27 1 Shifts and reﬂections of y ϭ . x

2 Ϫ2 (b) If we factor out Ϫ1fromthedenominator,then2ϪxϭxϪ2,so 2 2ϪxϭϪ2f͑xϪ2͒.Translatethegraphofftwo units to the right, reﬂect it through the x-axisandstretchitverticallybyafactorof2.Plottingafewpoints gives the graph shown in Figure 27b.

1 ϩ x 1 ϩ x (c) To relate to f ͑x͒, rewrite as x x 1 ϩ x 1 x 1 ϭ ϩ ϭ ϩ 1 ϭ f͑x͒ ϩ 1. x x x x Translate the graph of f one unit up, as shown in Figure 27c. ᭣

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(x͒ the zeros of the numerator? Suppose f͑x͒ ϭ and f is in reduced form (so that q͑x͒ p and q have no common zeros). Then if c is a zero of the numerator, we have p͑c͒ 0 f͑c͒ ϭ ϭ ϭ 0. That is, the zeros of a rational function are the zeros of q͑c͒ q͑c͒ the numerator. Collectively, we call the zeros of the numerator and the zeros of the denominator the set of cut points for a rational function. As in Chapter 1, cut points identify the location of possible sign changes. In addition, for rational functions, cut points in the numerator identify x-intercept points, cut points in the denominator correspond to actual breaks in the graph, at each vertical asymptote. Summing up, we have the following. Vertical asymptotes and intercepts p͑x͒ Suppose f ͑x͒ ϭ and that p and q have no common zeros. q͑x͒ Then there is a vertical asymptote at every zero of the denominator, and there is an x-intercept point at every zero of the numerator. pg184 [V] G2 5-36058 / HCG / Cannon & Elich clb 11-22-95 MP1

184 Chapter 3 Polynomial and Rational Functions

᭤EXAMPLE 3 Graphing a rational function Find the intercepts, cut points and asymptotes and sketch the graph of the rational function x ϩ 1 f͑x͒ ϭ . 4x͑x Ϫ 1͒ Solution The zeros of the denominator are 0 and 1, so the vertical asymptotes are the lines x ϭ 0(they-axis) and x ϭ 1. The other cut point is the single zero of the numerator, Ϫ1,andsowehaveonlyonex-intercept point, (Ϫ1, 0). Since the y-axis is a vertical asymptote, there is no y-intercept. A calculator graph in a decimal window [Ð 4, 4] by [Ð 3, 3] (Figure 28a) makes it appear that the graph ends near (Ϫ1, 0), but of course we (a) know that the domain of f consists of all real numbers except 0 and 1. We also know that the numerator, and hence f,changessignatxϭϪ1. Tracing along the curve y 1+x near (Ϫ1, 0) or zooming in near the same point veriﬁes that the graph crosses the y = 4x(xÐ1) 3 x-axis there, and the x-axis is a horizontal asymptote, as in Figure 28b. ᭣ 2 Horizontal and Slant Asymptotes 1 Inalloftheexampleswehavegraphedthusfar,thex-axis has been a horizontal x asymptote, with the exception of one translated graph. The general principle that Ð3Ð2 Ð1 1 2 3 Ð1 we list below is illustrated in all of these examples. What happens as x A ϱ or x A Ϫϱ Ð2 depends on the degrees of the numerator and denominator. Horizontal and slant asymptotes Ð3 axm ϩ ··· f ͑x͒ ϭ (b) bxn ϩ ··· FIGURE 28 If the degree of the denominator is larger ͑m Ͻ n͒, x ϩ 1 Graph of y ϭ 4x͑x Ϫ 1͒ then the x-axis is a horizontal asymptote. If the degrees are equal ͑m ϭ n͒, a .is a horizontal asymptote ؍ then the line y b Use long division to get additional information. If the degrees differ by 1 ͑m ϭ n ϩ 1͒, then there is a slant (oblique) asymptote. Use long division to ﬁnd an equation for the asymptote.

᭤EXAMPLE 4 Equal degrees Graph the rational function x 2 Ϫ 4 f͑x͒ ϭ . x 2 ϩ 2 Solution We note ﬁrst that f͑Ϫx͒ ϭ f͑x͒,sofis an even function; the graph is symmetric about the y-axis. Setting the numerator equal to 0, we ﬁnd that its zeros are Ϯ2, so the x-intercept points are (2, 0) and (Ϫ2, 0). The denominator has no real zeros, so the graph has no vertical asymptotes. Since f ͑0͒ ϭϪ2, the y-intercept point is (0, Ϫ2). pg185 [R] G1 5-36058 / HCG / Cannon & Elich jb 11-13-95 QC2

3.4 Rational Functions 185

y The degree of both numerator and denominator is 2, so m ϭ n. Thus the line a 1 y ϭ b ϭ 1 ϭ 1 is a horizontal asymptote. The graph is shown in Figure 29. Note 4 that without the boxed information above, it would be difficult to tell from a 3 (Ð 2, 0) y =1 2 calculator graph that the graph really does flatten out along the line y ϭ 1, al- 1 though increasing the x-range makes that conclusion very plausible. ᭣ x Ð4Ð3 Ð2Ð1Ð1 1 2 3 4 ᭤EXAMPLE 5 Equal degrees Graph the function, identifying all asymp- (2, 0) Ð2 totes and intercept points in exact form. (0, Ð 2) Ð3 x2 Ð4 y = 2 2 x ϩ 2x Ϫ 7 Ð4 x +2 g͑x͒ ϭ . x 2 Ϫ 2x Ϫ 3 FIGURE 29 Solution The zeros of the denominator are easy to find because the denominator factors: ͑x Ϫ 3͒͑x ϩ 1͒. There are zeros, and hence vertical asymptotes at x ϭϪ1and x=Ð1 xϭ3. The numerator has zeros at x ϭϪ1Ϯ2͙2, so we have two x-intercept (0, 7 ) points. g͑0͒ ϭ 7 ,sothey-intercept point is ͑0, 7͒. y=1 3 3 3 (1, 1) The degree of both numerator and denominator is 2, so m ϭ n. Thus the line a 1 y ϭ b ϭ 1 ϭ 1 is a horizontal asymptote. If we use long division, as suggested in (Ð 1 + 2 2, 0) the box above, we have (Ð 1 Ð 2 2, 0) 4x Ϫ 4 4͑x Ϫ 1͒ x =3 g͑x͒ ϭ 1 ϩ ϭ 1 ϩ . x 2 Ϫ 2x Ϫ 3 ͑x Ϫ 3͒͑x ϩ 1͒ Inthisform,itiseasytoseethatg͑x͒ϭ1 when x ϭ 1, so the graph crosses the [Ð 8, 8] by [Ð 6, 6] horizontal asymptote at the point (1, 1). Graphing in a decimal window doesn’t FIGURE 30 show much of the asymptotic behavior. To see a little more, we zoom out by a factor x 2 ϩ 2x Ϫ 7 of2togetthegraphshowninFigure30. ᭣ g͑x͒ ϭ x 2 Ϫ 2x Ϫ 3 Intersections of Graphs and Asymptotes Because graphs get so close to asymptotes it is sometimes difficult to look at a graph and tell whether or not it crosses an asymptote. Calculator graphs are particularly difficult to read in some areas where we most need detail. In Example 3 the graph crosses the horizontal asymptote at the x-intercept point (Ϫ1, 0) and in Example 5 the graph crosses the horizontal asymptote as well. When numerator and denominator have equal degrees, long division gives us a form from which we can use algebraic techniques to find intersections. On the other hand, no graph can cross a vertical asymptote. If x ϭ c is a vertical asymptote for a rational function f,then there would have to be a point ͑c, f͑c͒͒ on the graph, but vertical asymptotes occur at zeros of the denominator, where by definition, f is undefined.

Slant Asymptotes From the box above, slant (oblique) asymptotes occur when the degree of the numerator is 1 greater than the degree of the denominator, as in the next example.

᭤EXAMPLE 6 Degrees differ by 1 Graph the rational function x 2 ϩ 2x h͑x͒ ϭ . 2x ϩ 2 pg186 [V] G2 5-36058 / HCG / Cannon & Elich jb 11-13-95 QC2

186 Chapter 3 Polynomial and Rational Functions

Solution x͑x ϩ 2͒ Both numerator and denominator can be factored: . Thus the cut points 2͑x ϩ 1͒ are 0, Ϫ2, and Ϫ1. The ﬁrst two give us x-intercept points (0, 0) and (Ϫ2, 0); the zero in the denominator indicates a vertical asymptote at x ϭϪ1. A calculator graph in ͓Ϫ4, 4͔ ϫ ͓Ϫ4, 4͔ shows the graph near the vertical asymptote (Fig- ure 31a), but it isn’t clear what happens to the graph as we move further to the right or the left. If we zoom out, essentially looking at the graph from further away, say in the window ͓Ϫ10, 10͔ ϫ ͓Ϫ6, 6͔, the graph looks very much like a line, except near the vertical asymptote. What line does it approach? The slant asymptote.

y Oblique asymptote 4 x2 +2x x +1 y= 3y= 2x+2 2 2 (Ð 2, 0) 1 x (0, 0) Ð4 Ð3 Ð2 Ð1 1 234 Ð1 Ð2 Ð3 Ð4

[Ð4, 4] by [Ð 4, 4] (b)Oblique asymptote and 2 graph of y= x+2x (a) 2x+2

FIGURE 31 The numerator has degree 2 and denominator degree 1, so there is a slant asymptote. Dividing x ϩ 1intox2 ϩ2xusing long division, we get 1 1 h͑x͒ ϭ ͑x ϩ 1͒ Ϫ . 2 2͑x ϩ 1͒ The slant asymptote is the line y ϭ ͑x ϩ 1͒͞2.Addingthatlinetothegraphinthe ͓Ϫ10, 10͔ ϫ ͓Ϫ6, 6͔ window shows that the two graphs are indistinguishable except in the region near the vertical asymptote. As with horizontal asymptotes, we want to know whether the graph of y ϭ h͑x͒ intersects the oblique asymptote. Such an intersection would come from a solution to the equation h͑x͒ ϭ ͑x ϩ1͒͞2, which clearly has no solution. The graph is shown in Figure 31b. ᭣

y 2 Rational Functions Not Reduced y = x Ð 1 3 x Ð 1 p͑x͒ A rational function f͑x͒ ϭ is not in reduced form if there are common factors 2 q͑x͒ (1, 2) is in the numerator and denominator. To handle common zeros, remember that if p͑x͒ missing 0 1 and q͑x͒ have the same zero, say x ϭ c,thensince0 is not deﬁned, c is not in the domain of the function. In such a case the graph has a single point removed. x Consider the function Ð2Ð1 1 2 x 2 Ϫ 1 f͑x͒ ϭ . FIGURE 32 x Ϫ 1 pg187 [R] G1 5-36058 / HCG / Cannon & Elich cr 11-1-95 QC

3.4 Rational Functions 187

Factor the numerator. ͑x Ϫ 1͒͑x ϩ 1͒ f͑x͒ ϭ . x Ϫ 1 This is identical to the function g͑x͒ ϭ x ϩ 1 except when x is 1. When x is 1, g͑1͒ ϭ 2, but f͑1͒ is not deﬁned. Therefore, the graph of y ϭ f͑x͒ isthesameas the graph of y ϭ x ϩ 1withthepoint(1,2)removed(seeFigure32). TECHNOLOGY TIP ᭜ Missing points

In graphing nonreduced rational functions, we may or may not be able to see that a point is missing. Such a gap is visible on a calculator graph if and only if there is a pixel with x-coordinate corresponding to value where the 0 function has the form 0 . Thus for the example considered above, f ͑x͒ ϭ ͑x 2 Ϫ 1͒͑͞x Ϫ 1͒,inany window where there is a pixel for x ϭ 1, such as a decimal window, the graph makes the missing point quite apparent. If you then change the x-range by almost any small amount, you will no longer be able to see where the point is missing; the graph looks just like the line y ϭ x ϩ 1.

EXERCISES 3.4 Check Your Understanding Use a graph whenever you think it will be helpful. True or 3x 2 ϩ 1 Exercises 7–8 Suppose f͑x͒ ϭ . False. Give reasons. x 2 ϩ 1 1 1. If the graph of y ϭ is translated two units left, then 7. Thereisnovalueofxfor which f ͑x͒ ϭ 3. x 8. For every real number x, f ͑x͒ is in the interval ͓1, 3͒. 1 the resulting graph will be that of y ϭ . x 2 ϩ 100 x Ϫ 2 9. If f͑x͒ ϭ ,thenthegraphoffhas a local x 1 minimum point in the third quadrant. 2. If the graph of y ϭ is translated down one unit, x ϩ 2 2x 3 Ϫ 3x 2 ϩ 500 Ϫx Ϫ 1 10. The graph of f ͑x͒ ϭ 4 , has one zero then the resulting graph will be that of y ϭ . x ϩ 8x ϩ 50 x ϩ 2 and no vertical asymptotes. 1 3. If the graph of y ϭ is translated up one unit, then x ϩ 2 x ϩ 1 Develop Mastery the resulting graph will be that of y ϭ . x ϩ 2 1. The line x ϭ 1 is a vertical asymptote for the function x x 3 4. The line y ϭ is an asymptote to the graph of y ϭ f͑x͒ ϭ . 2 x 2 Ϫ 2x ϩ 1 x ϩ 1 A Ϫ . (a) To see what happens to the graph of f as x 1 , 2x ϩ 1 evaluate f at x ϭ 0.8, 0.9, 0.99, etc. For what values 5. The horizontal line y ϭϪ2 is an asymptote to the of x is f ͑x͒ greater than 100? Greater than 1000? ϩ 1 Ϫ 2x 2 (b) Repeat part (a) as x A 1 . graph of y ϭ . 5 ϩ 2x ϩ x 2 2. The line y ϭ 1 is a horizontal asymptote for the function x Ϫ 2 6. The graph of y ϭ has no vertical asymp- x 2 Ϫ x ϩ x 2 ϩ 2x 2 f͑x͒ ϭ . totes. x 2 ϩ 1 pg188 [V] G2 5-36058 / HCG / Cannon & Elich cr 11-1-95 QC

188 Chapter 3 Polynomial and Rational Functions

(a) Draw a graph of y ϭ f ͑x͒, Does the graph 2x x 2 ϩ 2 24. f ͑x͒ ϭ 25. f ͑x͒ ϭ approach the line y ϭ 1 from above or below as x 2 ϩ 1 x 2 ϩ 1 x A ϱ? 4 TRACE (b) Use and appropriate windows to find some 26. f ͑x͒ ϭ 2 values of x for which f ͑x͒ is within 0.01, 0.001, x Ϫ 3x Ϫ 4 of y ϭ 1. x Ϫ 2 27. f ͑x͒ ϭ (c) DothesameforxAϪϱ. x 3 Ϫ 3x 2 Ϫ 4x (d) Does the graph intersect the line y ϭ 1? If it 2x Ϫ 4 does, find the coordinates of the point of 28. f ͑x͒ ϭ x 3 Ϫ 3x 2 Ϫ 4x intersection algebraically. 3. Repeat Exercise 2 for the function Exercises 29–32 Increasing, Decreasing Determine x 2 ϩ 2x the intervals on which f is (a) increasing, (b) decreasing. f͑x͒ ϭ . x 2 ϩ 2 x 2 ϩ 4x ϩ 3 x 2 Ϫ 2x Ϫ 3 29. f ͑x͒ ϭ 30. f ͑x͒ ϭ 2 Exercises 4–9 Use translations, reflection, or stretching of 2x ϩ 4 x Ϫ 2x ϩ 3 1 x the graph of f ͑x͒ ϭ x to sketch a graph. See Example 2. Ϫ 1 4 31. f ͑x͒ ϭ 2 32. f ͑x͒ ϭ 2 1 1 2 x Ϫ x Ϫ 2 x ϩ 2x Ϫ 3 4. y ϭ 5. y ϭ 6. y ϭ x ϩ 2 1 Ϫ x x Ϫ 3 Exercises 33–36 Solution Set Find the solution set alge- 2 2x ϩ 1 x Ϫ 1 braically. 7. y ϭ 8. y ϭ 9. y ϭ 3 Ϫ x x x 2 4 x 2 Ϫ 6x ϩ 5 33. Ϫ Ϫ 6 ϭ 0 34. ϭ 3 2x 2 x x Ϫ 2 x 2 ϩ 3 Exercises 10–13 Related Graphs Graph f ͑x͒ ϭ 2 x ϩ 1 x 3 Ϫ 4x 3 and the given function g simultaneously. (a) Determine 35. ϭ x 2 ϩ 1 2 how the graphs are related and verify algebraically. 2x 4 14 (b) Find the coordinates of any intersection points (1 deci- 36. ϩ ϭ mal place). x ϩ 1 2x ϩ 1 2x 2 ϩ 3x ϩ 1 4x 2 ϩ 2 Ϫ2 10. g͑x͒ ϭ 11. g͑x͒ ϭ Exercises 37–40 Solution Set (a) Find the solution set x 2 ϩ 1 x 2 ϩ 1 algebraically. (b) Use a graph to support your answer. 2x 2 ϩ 4x ϩ 2 1 x Ϫ 3 12. g͑x͒ ϭ 37. Ͼ 2 38. Ͻ 0 x 2 ϩ 2x ϩ 2 x ϩ 1 x ϩ 2 2x 2 Ϫ 8x ϩ 8 2x 2 ϩ x Ϫ 3 x 2 Ϫ 3x Ϫ 4 13. g͑x͒ ϭ . 39. Յ 0 40. Ն 0 x 2 Ϫ 4x ϩ 5 x 2 ϩ 1 x 2 ϩ 2x ϩ 1 Exercises 14–21 Graph Rational Functions Sketch a Exercises 41–44 Not Reduced Sketch a graph of f. graph of f, identifying asymptotes and intercepts. (Hint: First express the function in reduced form; keep in 1 2 mind the domain.) 14. f ͑x͒ ϭ 15. f ͑x͒ ϭ x Ϫ 1 x ϩ 2 x x 3 41. f ͑x͒ ϭ 42. f ͑x͒ ϭ x 2x Ϫ 3 x 2 ϩ 2x x 2 Ϫ 2x 16. f ͑x͒ ϭ 17. f ͑x͒ ϭ x Ϫ 1 x ϩ 2 x 2 Ϫ 2x ϩ 1 x 2 ϩ 2x Ϫ 3 43. f ͑x͒ ϭ 44. f ͑x͒ ϭ 2x 2 x 2 x 2 ϩ 2x Ϫ 3 x 2 ϩ x Ϫ 2 18. f ͑x͒ ϭ 19. f ͑x͒ ϭ ͑x Ϫ 1͒2 x 2 Ϫ 4 Exercises 45–48 Find an equation for the horizontal x ϩ 2 x asymptote and find the coordinates of the points (if any) 20. f ͑x͒ ϭ 2 21. f ͑x͒ ϭ 2 x Ϫ 3x Ϫ 4 x Ϫ 2x ϩ 1 where the graph of y ϭ f ͑x͒ intersects the horizontal Exercises 22–28 Intercepts, Domain, Range Draw a asymptote. graph and use it to find (a) the x-intercept points, (b) the x 2 Ϫ 6 2x 2 Ϫ 2 domain of f, (c) the range of f. 45. f ͑x͒ ϭ 46. f ͑x͒ ϭ x 2 Ϫ 2x x 2 Ϫ 3x ϩ 2 2 2 22. f ͑x͒ ϭ 23. f ͑x͒ ϭ x 2 Ϫ x x 2 Ϫ x ͑x ϩ 2͒2 x 2 ϩ 1 47. f ͑x͒ ϭ 48. f ͑x͒ ϭ x 2 ϩ x ϩ 2 x 2 Ϫ x Ϫ 2 pg189 [R] G1 5-36058 / HCG / Cannon & Elich clb 11-22-95 MP1

3.4 Rational Functions 189

Exercises 49–52 Oblique Asymptotes The graph of f has 61.y 62. y an oblique asymptote. Find an equation for the asymptote and ﬁnd the coordinates of the points (if any) where the y=1 x =1 graphofy ϭ f͑x͒intersects the asymptote. Graph the func- x tion. x x 2 Ϫ 3 x 2 ϩ x Ϫ 2 49. f ͑x͒ ϭ 50. f ͑x͒ ϭ x Ϫ 1 x ϩ 1 2 2 2x ϩ 3x ϩ 1 x ϩ 3x Ϫ 2 63.y 64. y 51. f ͑x͒ ϭ 52. f ͑x͒ ϭ x x ϩ 1

y=1 Exercises 53–54 Minima Find the minimum value of f y=1 algebraically and check graphically. What value of x gives x the minimum value of f? x=2 x x 4 ϩ x 2 ϩ 1 x=Ð1 x=1 53. f ͑x͒ ϭ x 2 1 1 2 Hint: f͑x͒ ϭ x 2 ϩ 1 ϩ ϭ x Ϫ ϩ 3 x 2 ͩ xͪ 65.y 66. y x 4 ϩ 2x 2 ϩ 4 54. f ͑x͒ ϭ x 2 Hint: See Exercise 53. x x=Ð1 x=1 Exercises 55–56 Your Choice Giveaformulaforara- x tional function whose graph satisﬁes the given conditions. x=2 Check with a graph. 55. x-intercept point (2, 0), vertical asymptote x ϭϪ1, horizontal asymptote y ϭ 2. 56. x-intercept points (Ϫ2, 0), vertical asymptote x ϭ 1, 67.y 68. y horizontal asymptote y ϭ 2.

Exercises 57–58 Local Maxima Find the coordinates of the local maximum point(s) on the graph of f. x x 2 Ϫ 7x ϩ 16 x 2 ϩ x ϩ 1 57. f ͑x͒ ϭ 58. f ͑x͒ ϭ x=1 x Ϫ 3 x Exercises 59–60 Local Minima Find the coordinates of x=Ð1 the local minimum point(s) on the graph of f. y=x+1 x 2 ϩ 5x ϩ 7 x 2 Ϫ x ϩ 4 59. f ͑x͒ ϭ 60. f ͑x͒ ϭ x x ϩ 2 x x=2 Exercises 61–68 Match Functions Match the graph with the appropriate function from the following list. Check by graphing. x ϩ 1 1 (a) f ͑x͒ ϭ (b) f ͑x͒ ϭ x Ϫ 1 x Ϫ 1 Exercises 69–72 Solution Set Functions g and h are x ϩ 1 x 2 g͑x͒ (c) f ͑x͒ ϭ (d) f ͑x͒ ϭ given and function f is deﬁned by f͑x͒ ϭ . Find the ͑x Ϫ 1͒2 x 2 ϩ 1 h͑x͒ solution set for f͑x͒ Ͻ 0 algebraically. Draw a graph to 1 x 2 (e) f ͑x͒ ϭ (f) f ͑x͒ ϭ support your answer. (Hint: First show that h͑x͒ Ͼ 0 for 1 Ϫ x x 2 Ϫ x Ϫ 2 every value of x (draw a graph).) Why is the solution set for x x 3 f ͑x͒ Ͻ 0 the same as the solution set for g͑x͒ Ͻ 0? (g) f ͑x͒ ϭ (h) f ͑x͒ ϭ x 2 Ϫ x Ϫ 2 x 2 Ϫ x Ϫ 2 69. g͑x͒ ϭ x 2 Ϫ 2x Ϫ 3, h͑x͒ ϭ x 2 Ϫ 2x ϩ 3 pg190 [V] G2 5-36058 / HCG / Cannon & Elich cr 11-1-95 QC

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70. g͑x͒ ϭ x 3 Ϫ 2x 2 Ϫ 8x, h͑x͒ ϭ x 4 Ϫ 3x 2 ϩ 4 81. A cylindrical can is to contain 48 ounces (87 cubic 71. g͑x͒ ϭ x 3 ϩ 2x 2 Ϫ x Ϫ 2, h͑x͒ ϭ x 2 ϩ 2x ϩ 2 inches) of apple juice. If the can is to use the least amount of tin, what should the radius and the height be? 72. g͑x͒ ϭ 1 Ϫ x 3, h͑x͒ ϭ 1 ϩ x 2 82. Suppose x ounces of pure acid are added to 50 ounces 2x 3 ϩ 3x 2 ϩ x Ϫ 2 73. For f ͑x͒ ϭ 2 , of 40% solution of acid. Let u denote the concentration x (percent) of the resulting solution. (a) show that y ϭ 2x ϩ 3 is an oblique asymptote. (a) Express u as a function of x. (b) Draw graphs of y ϭ f͑x͒ and y ϭ 2x ϩ 3 simulta- (b) Why cannot u be 100 or greater? neously and see that the graph of y ϭ f ͑x͒ is ap- (c) What is the domain of this function? proaching the asymptote as x A ϱ. Is it approaching (d) How many ounces of acid must be added to get a from above or from below? 65% solution? Support your answer by drawing a (c) Does the graph of y ϭ f͑x͒ intersect the asymp- graph and finding the value of x that gives 65 for u. tote? If it does, find the point of intersection algebraically. 83. Find the positive number (2 decimal places) such that (d) Find some values of x for which the difference of the sum of its square and its reciprocal is a minimum. the two y values is less than 0.01. 84. A rectangular box is to have a base whose length is 74. Solve the same problem as in Exercise 72 except use twice its width, and whose volume is 2460 cubic inches. x 3 Ϫ 3x 2 ϩ 2x Ϫ 1 Of all such boxes, what are the dimensions (1 decimal f͑x͒ ϭ . First find an equation for place) of the one that will require the least amount of x 2 ϩ 1 material if the box has (a) atop?(b) no top? the oblique asymptote. 85. Solve the problem in Exercise 84 if the length of the 75. Show that if g and h are polynomial functions with no base is three times its width. common zeros and the degree of h is 3, then the g͑x͒ Exercises 86–89 Applied Minima function f͑x͒ ϭ (lowest terms) must have at least h͑x͒ 86. Arectangularprintedpageistohavemargins2inches one vertical asymptote. wide at the top and bottom and 1 inch wide on each of the two sides. If the page is to have 60 square inches of x Ϫ 3 76. (a) Function f͑x͒ ϭ is not a rational function, printed material, ͉ x ͉ ϩ 2 (a) What is the minimum possible area of the page? Why? (b) What are the dimensions of the page? (b) Drawagraphandseethatthegraphfhas two horizontal asymptotes. 87. A factory has a fixed daily overhead cost of $600. If it (c) By considering two cases, x Ն 0, and x Ͻ 0, find produces x units daily, then the cost for labor and mate- equations for the two horizontal asymptotes. rialsis3xdollars. The daily cost of equipment mainte- x 2 nance is dollars. Exercises 77–78 Intercepts and Asymptotes Does the 240000 graphoffhave(a) x-intercept points? (b) Any vertical (a) Find a function giving the total daily cost, c͑x͒, asymptotes? (Hint: Draw graphs of the numerator and de- when x units are produced. nominator separately.) (b) How many units should be produced each day to x 2 Ϫ 2x ϩ 5 x 3 ϩ x Ϫ 1 c͑x͒ 77. f ͑x͒ ϭ 78. f ͑x͒ ϭ minimize the cost per unit minimize ?(Hint: x 4 ϩ 3x Ϫ 4 x 2 Ϫ 3x ϩ 4 ͩ x ͪ x Ͼ 10000.) 79. Of all rectangles with an area of 160 square inches, what are the dimensions of the one having the smallest 88. The x-axis, y-axis and any line with negative slope pass- perimeter? ing through the point P͑3, 5͒ determine a triangle. Of all such triangles, determine the line for which the area 80. SolvetheprobleminExercise79forarectangleofarea of the triangle is a minimum. 240 square inches. 89. SolvetheprobleminExercise88ifPisthepoint(5,4). CHAPTER 3 REVIEW Test Your Understanding 3 True or False. Give reasons. 2. The equation x ϩ x ϩ 1 ϭ 0 has no positive roots. 3 1. F͑x͒ ϭ xϪ2 ϩ xϪ1 ϩ 1 is a polynomial function of 3. The equation x ϩ x Ϫ 1 ϭ 0 has no negative roots. degree Ϫ2. 4. The equation x3 ϩ x2 Ϫ 1 ϭ 0 has no positive roots.