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4 Appl inverse functions in the real world. Applications Functions that model real-life situations are frequentl epressed as formulas with letters that remind ou of the variable the represent. In finding the inverse of a function epressed b a formula, interchanging the letters could be ver confusing. Accordingl, we omit Step 2, keep the letters the same, and just solve the formula for the other variable. EXAMPLE 9 Water Pressure on Underwater Devices At the beginning of this section, we gave the formula for finding the water pressure p (in pounds per square inch) at a depth d (in feet) below the surface. This formula can be written as p = 5d Suppose the pressure gauge on a diving bell breaks and 11. shows a reading of 18 psi. How far below the surface was the bell when the gauge failed? SOLUTION We want to find the unknown depth in terms of the known pressure. This depth is given b the inverse of the function p = 5d To find the inverse, we solve the given 11. equation for d. 11p = 5d Original equation Multipl both sides b 11. Solve for d. Now we can use this formula to find the depth when the gauge read 18 psi. We let p = 18. d = 11p 5 d = 111182 5 d = 396 p = 5d 11 d = 11p 5 Depth from pressure equation Replace p with 18. The device was 396 feet below the surface when the gauge failed. Practice Problem 9 In Eample 9, suppose the pressure gauge showed a reading of 165 psi. Determine the depth of the bell when the gauge failed. SECTION 9 Eercises A EXERCISES Basic Skills and Concepts 1. If no horizontal line intersects the graph of a function f in more than one point, then f is a(n) function. 2. A function f is one-to-one when different -values correspond to. 3. If f12 = 3, then f -1 12 =. 4. The graphs of a function f and its inverse are smmetric in the line. f -1 5. True or False If a function f has an inverse, then the domain of the inverse function is the range of f. 6. True or False It is possible for a function to be its own inverse, that is, for f =f -1. 294
In Eercises 7 14, the graph of a function is given. Use the horizontal-line test to determine whether the function is one-to-one. 7. 8. 9. 1. 25. For f12 = 2-3, find each of the following. a. f132 b. f -1 132 c. 1f f -1 21192 d. 1f f -1 2152 26. For f12 = 3, find each of the following. a. f122 b. f -1 182 c. 1f f -1 21152 d. 1f -1 f21272 27. For f12 = 3 + 1, find each of the following. a. f112 b. f -1 122 c. 1f f -1 212692 28. For g12 = 23 2 3-1, find each of the following. a. g112 b. g -1 112 c. 1g -1 g211352 In Eercises 29 34, show that f and g are inverses of each other b verifing that f1g122 g1f122. 29. 3. 31. f12 = 3 + 1; g12 = - 1 3 f12 = 2-3; g12 = 2-3 f12 = 3 ; g12 = 23 32. f12 = 1 ; g12 = 1 33. f12 = - 1 1 + 2 ; g12 = + 2 1-11. 12. 34. f12 = 3 + 2 + 2 ; g12 = - 1-3 In Eercises 35 4, the graph of a function f is given. Sketch the graph of f 1. 35. 36. 13. 14. 37. 38. 4 2 In Eercises 15 24, assume that the function f is one-to-one. 15. If f122 = 7, find f -1 172. 16. If f -1 142 =-7, find f1-72. 17. If f1-12 = 2, find f -1 122. 18. If f -1 1-32 = 5, find f152. 19. If f1a2 =b, find f -1 1b2. 2. If f -1 1c2 =d, find f1d2. 21. Find 1f -1 f213372. 22. Find 1f f -1 2125p2. 23. Find 1f f -1 21-1582. 24. Find 1f -1 f2197282. 4 2 2 4 2 4 39. 4. 4 (4, 3) 2 4 2 2 4 2 4 (2, 3) ( 1, 5) 4 (2, 3) (6, 5) ( 1, 2) 4 2 4 2 4 ( 3, 5) 295
In Eercises 41 52, a. Determine whether the given function is a one-to-one function. b. If the function is one-to-one, find its inverse. c. Sketch the graph of the function and its inverse on the same coordinate aes. d. Give the domain and intercepts of each one-to-one function and its inverse function. 41. f12 = 15-3 42. g12 = 2 + 5 43. f12 = 24-2 44. f12 =-29-2 45. f12 =1 + 3 46. 47. g12 = 23+1 48. 49. f12 = 1 5. - 1, Z 1 51. f12 = 2 +1 + 1 52. 53. Find the domain and range of the function f of Eercise 33. 54. Find the domain and range of the function f of Eercise 34. In Eercises 55 58, assume that the given function is one-toone. Find the inverse of the function. Also find the domain and the range of the given function. + 1 55. f12 = 56. - 2, Z 2 1-2 57. f12 = 58. 1 +, Z-1 In Eercises 59 66, find the inverse of each function and sketch the graph of the function and its inverse on the same coordinate aes. 59. f12 =- 2, Ú 6. 61. f12 = ƒƒ, Ú 62. 63. f12 = 2 + 1, 64. 65. f12 =- 2 + 2, 66. f12 = 4-1 h12 = 23 1 - g12 = 1-1, Z f12 =-1 +1 + 2 g12 = + 2 + 1, Z-1 h12 = - 1-3, Z 3 g12 =- 2, g12 = ƒƒ, g12 = 2 + 5, Ú g12 =- 2-1, Ú B EXERCISES Appling the Concepts 67. Temperature scales. Scientists use the Kelvin temperature scale, in which the lowest possible temperature (called absolute zero) is K. (K denotes degrees Kelvin.) The function K1C2 =C+273 gives the relationship between the Kelvin temperature (K) and Celsius temperature (C). a. Find the inverse function of K1C2 =C+273. What does it represent? b. Use the inverse function from (a) to find the Celsius temperature corresponding to 3 K. c. A comfortable room temperature is 22 C. What is the corresponding Kelvin temperature? 68. Temperature scales. The boiling point of water is 373 K, or 212 F; the freezing point of water is 273 K, or 32 F. The relationship between Kelvin and Fahrenheit temperatures is linear. a. Write a linear function epressing K1F2 in terms of F. b. Find the inverse of the function in part (a). What does it mean? c. A normal human bod temperature is 98.6 F. What is the corresponding Kelvin temperature? 296 69. Composition of functions. Use Eercises 67 and 68 and the composition of functions to a. write a function that epresses F in terms of C. b. write a function that epresses C in terms of F. 7. Celsius and Fahrenheit temperatures. Show that the functions in (a) and (b) of Eercise 69 are inverse functions. 71. Currenc echange. Alisha went to Europe last summer. She discovered that when she echanged her U.S. dollars for euros, she received 25% fewer euros than the number of dollars she echanged. (She got 75 euros for ever 1 U.S. dollars.) When she returned to the United States, she got 25% more dollars than the number of euros she echanged. a. Write each conversion function. b. Show that in part (a), the two functions are not inverse functions. c. Does Alisha gain or lose mone after converting both was? 72. Hourl wages. Anwar is a short-order cook in a diner. He is paid $4 per hour plus 5% of all food sales per hour. His average hourl wage w in terms of the food sales of dollars is w = 4 +.5. a. Write the inverse function. What does it mean? b. Use the inverse function to estimate the hourl sales at the diner if Anwar averages $12 per hour. 73. Hourl wages. In Eercise 72, suppose in addition that Anwar is guaranteed a minimum wage of $7 per hour. a. Write a function epressing his hourl wage w in terms of food sales per hour. [Hint: Use a piecewise function.] b. Does the function in part (a) have an inverse? Eplain. c. If the answer in part (b) is es, find the inverse function. If the answer is no, restrict the domain so that the new function has an inverse. 74. Simple pendulum. If a pendulum is released at a certain point, the period is the time the pendulum takes to swing along its path and return to the point from which it was released. The period T (in seconds) of a simple pendulum is a function of its length l (in feet) and is given b T =11.112l. a. Find the inverse function. What does it mean? b. Use the inverse function to calculate the length of the pendulum assuming that its period is two seconds. c. The convention center in Portland, Oregon, has the longest pendulum in the United States. The pendulum s length is 9 feet. Find the period. l
75. Water suppl. Suppose is the height of the water above the opening at the base of a water tank. The velocit V of water that flows from the opening at the base is a function of and is given b V12 = 81. a. Find the inverse function. What does it mean? b. Use the inverse function to calculate the height of the water in the tank when the flow is (i) 3 feet per second and (ii) 2 feet per second. 76. Phsics. A projectile is fired from the origin over horizontal ground. Its altitude (in feet) is a function of its horizontal distance (in feet) and is given b = 64-2 2. a. Find the inverse function where the function is increasing. b. Use the inverse function to compute the horizontal distance when the altitude of the projectile is (i) 32 feet, (ii) 256 feet, and (iii) 512 feet. 77. Loan repament. Chris purchased a car at % interest for five ears. After making a down pament, she agreed to pa the remaining $36, in monthl paments of $6 per month for 6 months. a. What does the function f12 = 36, - 6 represent? b. Find the inverse of the function in part (a). What does the inverse function represent? c. Use the inverse function to find the number of months remaining to make paments if the balance due is $22,. 78. Demand function. A marketing surve finds that the number (in millions) of computer chips the market will purchase is a function of its price p (in dollars) and is estimated b = 8p 2-32p + 12, 6p 2. a. Find the inverse function. What does it mean? b. Use the inverse function to estimate the price of a chip if the demand is 118.5 million chips. C EXERCISES Beond the Basics In Eercises 79 and 8, show that f and g are inverses of each other b verifing that f1g122 g1f122. 79. 8. 1 3 4 f12 3 5 2 1 2 3 4 f12-2 -3 1 81. Let f12 = 24-2. a. Sketch the graph of =f12. b. Is f one-to-one? c. Find the domain and the range of f. 3 5 2 g12 1 3 4-2 -3 1 g12 1 2 3 4 82. Let f12 = Œœ + 2 where Œœ is the greatest integer Œœ - 2, function. a. Find the domain of f. b. Is f one-to-one? 83. Let f12 = + 1 = 1-1 + 1. a. Show that f is one-to-one. b. Find the inverse of f. c. Find the domain and the range of f. 84. Let g12 = 21-2, 1. a. Show that g is one-to-one. b. Find the inverse of g. c. Find the domain and the range of g. 85. Let P13, 72 and Q17, 32 be two points in the plane. a. Show that the midpoint M of the line segment PQ lies on the line =. b. Show that the line = is the perpendicular bisector of the line segment PQ; that is, show that the line = contains the midpoint found in (a) and makes a right angle with PQ. You will have shown that P and Q are smmetric about the line =. 86. Use the procedure outlined in Eercise 85 to show that the points 1a, b2 and 1b, a2 are smmetric about the line =. (See Figure 9.) 87. The graph of f12 = 3 is given in Section 6. a. Sketch the graph of g12 =1-12 3 + 2 b using transformations of the graph of f. b. Find g -1 12. c. Sketch the graph of g -1 12 b reflecting the graph of g in part (a) about the line =. 297
88. Inverse of composition of functions. We show that if and g have inverses, then 1f g2-1 =g -1 f -1. 1Note the order.2 Let f12 = 2-1 and g12 = 3 + 4. a. Find the following: (i) f -1 12 (ii) g -1 12 (iii) 1f g212 (iv) 1g f212 (v) 1f g2-1 12 (vi) 1g f2-1 12 (vii) 1f -1 g -1 212 (viii) 1g -1 f -1 212 b. From part (a), conclude that (i) 1f g2-1 =g -1 f -1. (ii) 1g f2-1 =f -1 g -1. 89. Repeat Eercise 88 for f12 = 2 + 3 and g12 = 3-1. 9. Show that f12 = 2>3, is a one-to-one function. Find f -1 12 and verif that f -1 1f122 = for ever in the domain of f. f Critical Thinking 91. Does ever odd function have an inverse? Eplain. 92. Is there an even function that has an inverse? Eplain. 93. Does ever increasing or decreasing function have an inverse? Eplain. 94. A relation R is a set of ordered pairs 1, 2. The inverse of R is the set of ordered pairs 1, 2. a. Give an eample of a function whose inverse relation is not a function. b. Give an eample of a relation R whose inverse is a function. SUMMARY Definitions, Concepts, and Formulas 1 The Coordinate Plane i. Ordered pair. A pair of numbers in which the order is specified is called an ordered pair of numbers. ii. Distance formula. The distance between two points P1 1, 1 2 and Q1 2, 2 2, denoted b d1p, Q2, is given b d1p, Q2 = 21 2-1 2 2 +1 2-1 2 2. iii. Midpoint formula. The coordinates of the midpoint M1, 2 of the line segment joining P1 1, 1 2 and Q1 2, 2 2 are given b 1, 2 = a 1 + 2 2, 1 + 2 b. 2 2 Graphs of Equations i. The graph of an equation in two variables, sa, and, is the set of all ordered pairs 1a, b2 in the coordinate plane that satisfies the given equation. A graph of an equation, then, is a picture of its solution set. ii. Sketching the graph of an equation b plotting points Step 1 Step 2 Make a representative table of solutions of the equation. Plot the solutions in Step 1 as ordered pairs in the coordinate plane. Step 3 Connect the representative solutions in Step 2 with a smooth curve. 298