Rational Functions Dr. Philippe B. Laval Kennesaw State University Abstract This handout is a quick review of rational functions. It also contains a set of problems to guide the student through understanding rational functions. This is to be turned in on Monday March 25. 1 Definition and Examples Definition 1 A rational function is a function of the form r (x) = P (x) Q (x) where P and Q are polynomials. We assume P and Q have no factors in common. We notice immediately that r is not defined for every real number. The domain of r is the set of real numbers which are not zeros of Q. The notation x a + means that x is approaching a and is larger than a. In other words, x is approaching a from the right. Similarly, the notation x a means that x is approaching a and is less than a. In other words, x is approaching a from the left. Definition 2 (asymptote) Consider the function y = f (x). 1. The line x = a is a vertical asymptote for f if y or y as x a + or x a. 2. The line y = b is a horizontal asymptote for f if y b as x or x. A rational function has a vertical asymptote wherever its denominator is zero but its numerator is not zero. In other words, the line x = a isavertical asymptote for some rational function, if and only if a is a zero of the denominator but not of the numerator. The graphs below illustrate the relationship between a function and its asymptote. In the graph below, the line x =2is a vertical asymptote. Here, we have that y as x 2 and y as x 2 + 1
In the graph below, the line x =2is a vertical asymptote. Here, we have that y as x 2 and y as x 2 + In the graph below, the line x =2is a vertical asymptote. Here, we have that y as x 2 as x 2 + 2
In the graph below, the line x =2is a vertical asymptote. Here, we have that y as x 2 as x 2 + The four graphs above illustrate the four possibilities for the behavior of a 3
function near its vertical asymptote. The graphs below shows the behavior of a function with respect to its horizontal asymptote. The horizontal asymptote represents the behavior of a function when x ±. In the graph below, the line y =2is a horizontal asymptote. Can you think of other possibilities for the position of a function with respect to its horizontal asymptote? When a function y = f (x) has a vertical asymptote at x = a, it means that it is not defined at x = a since a is a zero of the denominator. Therefore, we cannot evaluate f (a). According to the definition of an asymptote, we know that y ± as x ±. The question is to know whether y is approaching or. Inordertofind the answer, we evaluate the function at values of x closer and closer to a to see if we can detect a pattern. For example, consider the function y = f (x) = 1. We see that f is not defined at x =1.Tofind x 1 what y is approaching as x 1 + and x 1, we plug into the function values of x close to 1. Which values we try is not too important. What matters is that we try values of x close to 1 and getting closer and closer. To see what happens to y as x 1 +, we might try 1.1, 1.01, 1.001, 1.0001. The table below helps us with the answer. x 1.1 1.01 1.001 1.0001 y 10 100 1000 10000 We can see that as x getscloserto1 from the right, y is getting larger and larger. This suggests that y as x 1 +. To see what happens to y as x 1,wemighttry0.9, 0.99, 0.999, 0.9999. The table below helps us with the answer. 4
x 0.9 0.99 0.999 0.9999 y 10 100 1000 10000 We can see that as x gets closer to 1 from the left, y is getting larger and larger in absolute value, but it is also negative. This suggests that y as x 1. Using the concepts above, answer the questions below. 1.1 Group work Example 3 Consider f (x) = 1 x. answers. Answer the questions below, explain your 1. What is the domain of f? 2. With the help of a table of values, determine the value f approaches when x approaches. An equivalent way of asking this is: find what f approaches when x. 3. Same question when x 4. Obviously, f is not defined at 0. We wish to study the behavior of f as x approaches 0. x can either approach 0 from the right (x is larger than 0 and approaches 0), in this case we write x 0 +. x can also approach 0 from the left (x is smaller than 0 and approaches 0), in this case we write x 0. Using a table of values, determine what f approaches when x 0 +,whenx 0 5. Sketch the graph of f and its asymptotes. Example 4 Consider g (x) = 1. Answer the questions below, explain your x 2 answers. 1. What is the domain of g? 2. With the help of a table of values, determine the value g approaches when x approaches. An equivalent way of asking this is: find what g approaches when x. 3. Same question when x 4. Determine what g approaches when x 2 +,whenx 2 5. Sketch the graph of g and of its asymptotes. 6. Questions 1-5 can also be answered by considering g as a transformation of f (see 2.4). Show how you would do this by first determining which transformation would produce g from f. 5
Conclusion 5 Theimportantresulttorememberisthatforpositiverealnumbers, 1 big number = small number and 1 = big number. Another small number way of saying this is: A fraction whose denominator approaches ± and whose numerator approached some constant, will approach 0. A fraction whose denominator approaches 0, whose numerator does not approach 0, will go to infinity. It will be if the fraction is positive. It will be if the fraction is negative. 2 Graphing a Rational Function To sketch the graph of a rational function, the following information has to be determined. domain x and y-intercepts vertical asymptotes and behavior of the function near the vertical asymptotes horizontal asymptotes end behavior We look at each of these in details through an example. We will use the function f (x) = x 2 x 2 in the subsections which follow to illustrate the concepts 1 outlined above. 2.1 Finding the Domain Thedomainisthesetofrealnumbersforwhichafunctionisdefined. In the case of a rational function, it is the set of all real numbers which are not zeros of the denominator. To find the domain of a real function, we must then look for the real zeros of its denominator. This is when the tools learned in the previous sections will be useful. In our case, it is easy. The denominator of f is x 2 1=(x 1) (x +1). Its zeros are ±1. Thus, the domain of f is the set of all real numbers except ±1. 2.2 Finding the Intercepts x-intercepts. The x-intercepts are the values of x for which f (x) =0. A rational function is 0 if and only if its numerator is 0. Thus, finding the x-intercepts of a rational function is equivalent to finding the real zeros of its numerator. In our case, the x-intercept is x =2. 6
y-intercept. The y-intercept is the point where the graph intersects with the y-axis. Thus, to find it, we set x =0. In other words, we compute f (0). f (0) = 0 2 0 2 1 = 2 They-interceptisy =2. 2.3 Finding the Vertical Asymptotes A rational function will have a vertical asymptote at the zeros of its denominator (as long as they are not also zeros of the numerator). We still need to study the behavior of the function on either side of the asymptote. That is, does the functiongoto or? Therefore, f has two vertical asymptotes. They are the lines x = 1 and x =1. Now, we need to determine what f approaches as x 1, x 1 +, x 1 and x 1 +. Weknowitwillbeeither or since we have a vertical asymptote there. To find out which one it is, we simply need to know the sign of the fraction there. If the fraction is positive, it will approach. Otherwise, it will approach. This is best done with a table of value. For example, if x 1,thenx is close to 1 and less than 1. A possible value is 1.01. We evaluate f ( 1.01) = 149. 75. Thus, as x 1, f (x). If x 1 +, a possible value for x is.99. Since f (.99) = 150. 25, we conclude that as x 1 +, f (x). Similarly, if x 1,a possible value for x is..99. Sincef (.99) = 50. 754, weconcludethatasx 1, f (x). Finally, if x 1 +, a possible value for x is 1.01. Since f (1.01) = 49. 254, we conclude that as x 1 +, f (x). 2.4 Finding the Horizontal Asymptotes. The horizontal asymptotes are found by studying the behavior of the rational function as x ±. We first derive the general result. Consider the rational function r (x) = a nx n + a n 1 x n 1 +... + a 1 x + a 0 b m x m + b m 1 x m 1 +... + b 1 x + b 0 Recall that when x ±, a polynomial behaves like its leading term. Therefore, r (x) will behave like a nx n. Thus, we have: b m xm Proposition 6 Let r (x) = a nx n + a n 1 x n 1 +... + a 1 x + a 0 b m x m + b m 1 x m 1 +... + b 1 x + b 0. Then, we have the following: 1. If m>n,thenr has horizontal asymptote y =0. In this case, the end behavior is determined by the asymptote, no further study is needed. 7
2. If m = n, thenr has horizontal asymptote y = a n b m. In this case, the end behavior is determined by the asymptote, no further study is needed. 3. If n>m,thenr has no horizontal asymptote. Further study is needed for the end behavior. Since f falls in case 1, y =0is the horizontal asymptote. 2.5 End Behavior Like for polynomials, we wish to know the behavior of the rational function when x ±. We saw earlier that when the function has a horizontal asymptote, then we know the answer. The end behavior is given by the horizontal asymptote. This part only concerns the case when there is no horizontal asymptote, that is when the degree of the numerator is higher that the degree of the denominator. To use the notation of the previous subsection, n>m. Let r (x) = P (x). Since degree of P>degree of Q, we can perform long division. Q (x) The division algorithm tells us that there exists polynomials D (x) and R (x) such that P (x) =Q (x) D (x)+r(x). Therefore, r (x) = P (x) Q (x) Q (x) D (x)+r(x) = Q (x) Q (x) D (x) = + R (x) Q (x) Q (x) = D (x)+ R (x) Q (x) where D (x) is a polynomial, and the degree of R is strictly less than the degree of Q. Then, as x, R (x) 0. Therefore, r (x) will behave like D (x) for Q (x) large values of x. Definition 7 If D (x) as above is linear, say D (x) =ax + b, thenax + b is called a slant or an oblique asymptote for r (x). Example 8 Find the end behavior of r (x) = x3 1 x +1 If we perform long division, we find that r (x) = x3 1 x +1 = x 2 x +1 2 x +1 8
Therefore, r (x) will behave like x 2 x +1 when x ±. This can be verified by graphing both functions. In the graph below, r (x) is in blue, x 2 x +1 is in red. We can see that for large (not so large in fact), the blue graph and the red graph are very close, and getting closer as x increases. Figure 1: 2.6 ProblemstoTurnin Example 9 For each function below, do the following: find the domain find the intercepts find the asymptotes find the end behavior sketch the graph and the asymptotes 3x +3 1. r 1 (x) = x 3 2. r 2 (x) = x2 2 x +1 9
3. r 3 (x) = x2 x 6 x 2 +3x 4. r 4 (x) = 6x4 x 2 3 Example 10 Same question with f (x) = 3x2 3x 6. What do you notice? x 2 What did this teach you? Example 11 Find a rational function which has vertical asymptotes x = 1 and x =2, and horizontal asymptote y =1. Example 12 Do # 60, 61, 62 and 65 on pages 275, 276 10