Lesson 10 - Practice Problems Section 10.1: Characteristics of Rational Functions 1. Complete the table below. Function Domain a) f ( x) = 4x 6 2x b) f ( x) = 8x + 2 3x 9 c) s( t ) = 6t + 4 t d) p( t ) = t 12t 6 e) f ( x) = 2x 1 4x 2 f) g( x) = 3x 4 2x + 3 g) f ( x ) = 8 3x 417
Section 10.2: Simplifying Rational Expressions 2. Simplify the following rational expressions. Begin by determining the domain. a) 4x 6 2x b) 5x 20x 15 c) 3x + 5 12x + 20 d) 6x 8 3x 4 e) 12x + 21 8x + 14 f) 6x 5 5 6x g) x + 2 3(x + 2)(x 5) h) x 3 x 2 5x + 6 i) 6x 3x 2 6x j) x 2 2x + 1 x 2 + 2x 1 418
Section 10.3: Operations with Rational Expressions 3. Perform the indicated operation. Begin by determining the domain. a) 3 b) 5 x + 5 x 3x + 7 3x c) 4 d) 6 x 5 2x 5x + 1 3x e) 3 x + 1 + 5 x 2 f) 4x x 3 5 6x g) x + 3 x + 7 + 5 2x x 2 h) 3 x + 1 5 x i) x 4x 12 x 3 x + 1 j) 2x 2 3x 3x + 12 x 1 419
4. Perform the indicated operation. Begin by determining the domain. a) 3x 2x + 6 24 x + 3 b) x + 5 x 3 4x +20 2x 6 c) 3 x + 1 12 x + 1 d) 3x 12 x + 2 x 4 x + 2 5. Perform the indicated operation. Begin by determining the domain. a) 1 x + 2 y b) x y + y x c) 1 x + 2 y x y 2 x d) 3 x x y y x 2 y 420
Section 10.4: Solving Rational Equations 6. Solve each of the following equations by graphing. Round answer(s) to two decimal places as needed. a) 4 = 3x x 1 b) 3 = 2 + 5 x 2 c) Solution: x + 2 = 2 x 2 4 Solution: d) 3 = x + 1 x Solution: Solution: e) 5 x + 4 = 12 13 f) 2 x = 1 Solution: Solution: 421
7. Solve each of the following rational equations algebraically. Check your work by substituting the value(s) back into the equation or by graphing. a) 4 = 3 x 6 b) 4 x + 4 = 6 x 2 4 2x c) 3 = 3x + 2 4 d) 6 = 2 + 3 x 5 e) x + 4 = 4 x f) 1 x 3 = x + 3 5 422
8. Graph the given rational function on the grid below and complete the table marking DNE for any items that do not exist. f ( x) = 3 x 2 Domain Intercept(s) Intercept 423
9. Graph the given rational function on the grid below and complete the table marking DNE for any items that do not exist. g( x) = 3x + 4 x 3 Domain Intercept(s) Intercept 424
10. Graph the given rational function on the grid below and complete the table marking DNE for any items that do not exist. h( x) = x + 7 4 2x Domain Intercept(s) Intercept 425
11. Graph the given rational function on the grid below and complete the table marking DNE for any items that do not exist. f ( x) = 2x 1 x 2 2x Domain s Intercept(s) Intercept 426
12. Given f x a) Find f 1 ( ) = 3, determine the following: x 2 ( ) b) Findxsothatf x ( ) = 6 13. Given a) Find h 2 h( x) = 3x + 4, determine the following: x + 3 ( ) b) Findxsothath ( x ) = 16 7 14. Given f x a) Find f 1 ( ) = x + 7, determine the following: 4 2x ( ) b) Findxsothatf x ( ) = 3 15. Given ( ) = 2x 1 x( x 2) ( ) b) g x a) Find g 3, determine the following: Findxsothatg ( x) = 7 8 427
Section 10.5: Applications of Rational Functions 16. Mr. Sculley decides to make and sell Left Handed Smoke Shifters as a side business. The fixed cost to run his business is $250 per month and the cost to produce each Smoke Shifter averages $8. The Smoke Shifters will sell for $19.95. The function below gives the average cost (in dollars) per smoke shifter when x smoke shifters are produced. A ( x 8x + 250 ) = x a) Determine A (1), and write a sentence explaining the meaning of your answer. b) Complete the table below. x 1 2 3 4 5 A ( x ) c) Determine A (10), and write a sentence explaining the meaning of your answer. d) How many smoke shifters must be produced in order to reduce the average cost to $15 each? Write your answer in a complete sentence. e) Give the equation of the horizontal asymptote of A x ( ), and write a sentence explaining its significance in this situation. 428
17. You and your friends are heading out to San Deigo on a road trip. From Scottsdale, the trip is 373 miles. Answer the following questions based upon this situation. a) Use the relationship, Distance = Rate times Time or d = rt, to write a rational function T r ( ) that has the rate of travel, r (in mph), as its input and the time of travel (in hours) as its output. b) Provide a rough but accurate sketch of the graph in the space below. Label your horizontal and vertical axes. You only need to graph the first quadrant information. Indicate the graphing window you chose. Xmin = Xmax = Ymin = Ymax = c) According to Google, the trip should take 5 hours and 45 minutes (5.75 hours). Determine your average rate of travel if the trip takes only 5 hours. Write your answer in a complete sentence. d) Determine the horizontal asymptote for T r ( ), and write a sentence explaining its significance in this situation. 429
18. Harkins Theaters offers $1.50 soft drink refills every time you bring your Harkins Loyalty Cup to the theater. You can purchase the Loyalty Cup (filled) for $6.50. The function C ( x 1.5x +6.5 ) = gives the average cost (in dollars) per refill with the Loyalty x Cup, where x is the number of soft drink refills purchased. a) Determine C (1), and write a sentence explaining the meaning of your answer. b) Complete the table below. x 1 2 3 4 5 C ( x ) c) How many refills must you purchase in order to reduce the average cost to $2 per refill? Write your answer in a complete sentence. d) Give the equation of the horizontal asymptote of C x ( ), and write a sentence explaining its significance in this situation. 430