Rational Functions. Example 1. Find the domain of each rational function. 1. f(x) = 1 x f(x) = 2x + 3 x 2 4

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1 Rational Functions Definition 1. If p(x) and q(x) are polynomials with no common factor and f(x) = p(x) for q(x) 0, then f(x) is called a rational function. q(x) Example 1. Find the domain of each rational function. 1. f(x) = 1 x f(x) = 2x + 3 x 2 4 Chapter 1: 1.4 Rational Functions 1

2 Consider f(x) = 1 whose domain is the set of all real numbers x except x 0. We will determine the behavior of f near this excluded value. The values of f(x) to the left and right of x = 0 are given in the following tables. x f(x) x f(x) From the table, note that as x approaches 0 from the left, f(x) decreases without bound. In contrast, as x approaches 0 from the right, f(x) increases without bound. The graph of f is shown in Figure 4.1. Chapter 1: 1.4 Rational Functions 2

3 Figure 4.1 Chapter 1: 1.4 Rational Functions 3

4 From the graph of f(x) = 1, the function values approaches 0 as x x approaches negative infinity, that is, f(x) 0 as x. Also, the function values approaches negative infinity as x approaches zero from the left, that is, f(x) as x 0. Similarly, f(x) as x 0 + and f(x) 0 as x. Definition 2. The line x = a is a vertical asymptote of the graph of f if f(x) or f(x) as x a either from the right or the left. The line y = b is a horizontal asymptote of the graph of f if f(x) b as x or x. Chapter 1: 1.4 Rational Functions 4

5 Vertical and Horizontal Asymptotes of a Rational Functions Let f be the rational function f(x) = p(x) q(x) = a nx n + a n 1 x n a 1 x + a 0 b m x m + b m 1 x m b 1 x + b 0 where a n 0 and b m 0, p(x) and q(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of q(x). 2. The graph of f has at most one horizontal asymptote which varies with the degree of p(x) and q(x). (a) If n < m, the graph of f has the line y = 0 (the x horizontal asymptote. axis) as a Chapter 1: 1.4 Rational Functions 5

6 (b) If n = m, the graph of f has the line y = a n b m as a horizontal asymptote, where a n is the leading coefficient of the numerator and b m is the leading coefficient of the denominator. (c) If n > m, the graph of f has no horizontal asymptote. Chapter 1: 1.4 Rational Functions 6

7 Example 2. Find the horizontal and vertical asymptotes for each rational function. 1. f(x) = x + 1 x f(x) = 3x + 1 x + 2 Chapter 1: 1.4 Rational Functions 7

8 Let f be the rational function Oblique Asymptotes f(x) = p(x) q(x) = a nx n + a n 1 x n a 1 x + a 0 b m x m + b m 1 x m b 1 x + b 0 where a n 0 and b m 0, p(x) and q(x) have no common factors. n = m + 1, use division to rewrite the function as If quotient + remainder divisor The graph of f has the line y = quotient as oblique asymptote. Chapter 1: 1.4 Rational Functions 8

9 Example 3. Determine all the asymptotes of f(x) = x2 + 3 x + 4. Example 4. Sketch the graph of the rational functions given in Example 2 and state the domain and range. 1. f(x) = x + 1 x f(x) = 3x + 1 x + 2 Example 5. Sketch the graph of the function f(x) = x2 + 3 x + 4 Example 3 and state the domain and range. given in Chapter 1: 1.4 Rational Functions 9

10 Guidelines for Graphing Rational Functions Let f(x) = p(x) where p(x) and q(x) are polynomials with no common q(x) factors where q(x) Find and plot the y-intercept, if any, by evaluating f(0). 2. Find the zeros of the numerator, if any, by setting the numerator equal to zero. Then plot the x-intercepts. 3. Find the zeros of the denominator, if any,by setting the denominator equal to zero. Then sketch the corresponding vertical asymptotes using dashed vertical lines. 4. Find and sketch any other asymptotes of the graph using dashed lines. Chapter 1: 1.4 Rational Functions 10

11 5. Plot at least one point between and one point beyond each x-intercept and vertical asymptote. 6. Use smooth curves to complete the graph between and beyond the vertical asymptotes, excluding any points where f is not defined. Chapter 1: 1.4 Rational Functions 11

12 Example 6. Sketch the graph of f(x) = x + 3 2x range. and state the domain and Example 7. Sketch the graph of f(x) = and range. x x 2 x 6 and state the domain Example 8. Sketch the graph of f(x) = x2 2x 3 x 1 and range. and state the domain Chapter 1: 1.4 Rational Functions 12

13 In business, if a cost function C(x) represents the cost of producing x units, then the average cost function AC(x) is the cost divided by the number of units produced. Hence, AC(x) = C(x) x Example 9. A manufacturer has determined that the cost in pesos of producing hand towels is given by C(x) = 0.01x 2 3x , where x represents the number of hand towels produced daily. Determine the average cost of producing 100, 200 and 300 hand towels per day. Example 10. If a scuba diver goes to depth greater than 33 feet, the function T (d) = 1700 gives the maximum time a diver can remain down d 33 and still surface at a steady rate with no decompression stops. In this function, T (d) represents the dive time in minutes, and d represents the depth in feet. If a diver plans a 30-minute dive,what is the maximum depth the diver can go without decompression stops on the way back-up? Chapter 1: 1.4 Rational Functions 13

14 A rational equation is an equation containing at least one rational expression. Rational expressions typically contain a variable in the denominator. For this reason, we need to take note that the denominator must not be zero by considering necessary domain restrictions and checking our solutions. Solving rational equations involves clearing fractions by multiplying both sides of the equation by the least common denominator (LCD). Example 11. Solve 2x 3x + 1 = 1 x 5 4(x 1) 3x 2 14x 5. Chapter 1: 1.4 Rational Functions 14

15 A rational inequality is a mathematical statement that relates a rational expression as either less than or greater than another. In solving rational inequalities we need to consider the domain restrictions, the zeros of the denominator. In addition, we also need to consider the zeros of numerator. The zeros of both the numerator and the denominator are called the critical numbers. Chapter 1: 1.4 Rational Functions 15

16 Solving Rational Inequalities 1. Write the inequality in the standard form. One side must be zero and the other side is a single rational expression which we denote by f(x). 2. Find the critical numbers. These are the zeros of f(x) and the excluded values of f(x). Set the numerator and denominator of f(x) equal to zero and solve. 3. Divide the number line into intervals according to the critical numbers obtained in Step Choose a test value in each interval in Step 3, and construct a table. Substitute the test value to f(x) and determine the sign of the resulting answer. The sign of this answer (positive or negative) will be the sign of the entire interval. You can check using a different number from the same interval if you want to verify your answer. Chapter 1: 1.4 Rational Functions 16

17 5. Use the table in step 4 to determine which intervals satisfy the inequality. If the inequality is of the form f(x) < 0 or f(x) 0 then all of the intervals with the negative sign are solutions. In addition, the zeros of f(x) are part of the solution if f(x) 0. On the other hand, if the inequality is of the form f(x) > 0 or f(x) 0 then all of the intervals with the positive sign are solutions. In addition, the zeros of f(x) are part of the solution if f(x) 0. Chapter 1: 1.4 Rational Functions 17

18 Example 12. Solve 5 x 4 < 3 x + 1. Example 13. Solve x2 3x 10 1 x 2. Chapter 1: 1.4 Rational Functions 18

19 Exercises Sketch the graph of the rational function and state the domain and range. As sketching aids, check for intercepts and asymptotes. Use a graphing utility to verify your graph. 1. f(x) = 1 x 6 4. f(x) = 2 x 2 5x f(x) = 1 2x x 5. f(x) = 1 (x 2) 2 3. f(x) = x x f(x) = 3x + 8 x 2 Chapter 1: 1.4 Rational Functions 19

20 7. f(x) = 2x2 x 3 9. f(x) = 1 x 2 2x 8. f(x) = 3x2 + 2 x f(x) = x2 + 2x + 1 x Chapter 1: 1.4 Rational Functions 20

21 Solve each equation k 2 = 1 3k 2 1 k 6. 2 x x = x x t + t 8 = b b 5 b 1 = 2b + 5 b x = x 34 2x x 5 3x + 34x4 = 3x 2x c + 6 4c c 2 = c + 4 2c = 1 1 z + z z y = 6 5y y 1 = 7 2y + 3 y + 1 Chapter 1: 1.4 Rational Functions 21

22 Solve and graph. notation. Express answers in both inequality and interval 1. x2 + 5x x x > x 4 x 2 + 2x 0 (x + 1) 2 x 2 + 2x x + 1 x x 8 x x2 x 12 x x x x < x 3 2 x + 2 Chapter 1: 1.4 Rational Functions 22

23 Answer each question. 1. A manufacturer has determined that the cost in pesos of producing shirts is given by C(x) = 0.5x 2 x , where x represents the number of shirts produced daily. Determine the average cost of producing 50, 100, and 150 shirts per day. 2. A manufacturer has determined that the cost in pesos of producing portable speakers is given by the function C(x) = 3x(x 100) + 32, 000, where x represents the number of speakers produced in a week. Determine the average cost per speaker if 50 are produced in a week. 3. A rectangular region of length x and width y has an area of 400 square meters. (a) Write the width y as a function of x. (b) Determine the domain of the function based on the physical conditions of the problem. Chapter 1: 1.4 Rational Functions 23

24 (c) Sketch a graph of the function and determine the width of the rectangle when x = 25 meters. 4. The cost in pesos of an environmental cleanup is given by the function 30, 000p C(p) =, where p represents the percentage of the area to be 1 p cleaned up (0 p < 1). Use the function to determine the cost of cleaning up 50area. 5. When a successful new software is first introduced, the weekly sales generally increase rapidly for a period of time and then begin to decrease. Suppose that the weekly sales S (in thousands of units) t weeks after the software is introduced are given by S = 200t t When will sales be 8 thousand units per week or more? Chapter 1: 1.4 Rational Functions 24

Introduction. A rational function is a quotient of polynomial functions. It can be written in the form

Introduction. A rational function is a quotient of polynomial functions. It can be written in the form RATIONAL FUNCTIONS Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 2 In general,