For Thought. 2.1 Exercises 80 CHAPTER 2 FUNCTIONS AND GRAPHS

Precalculus Functions and Graps 4t Edition Dugopolski SOLUTIONS MANUAL Full download at: ttps://testbankreal.com/download/precalculus-functions-and-graps-4t-editiondugopolski-solutions-manual/ Precalculus Functions and Graps 4t Edition Dugopolski TEST BANK Full download at: ttps://testbankreal.com/download/precalculus-functions-and-graps-4t-editiondugopolski-test-bank/ 80 CHAPTER FUNCTIONS AND GRAPHS For Tougt. False, since {(, ), (, )} is not a function.. False, since f (5) is not defined.. True 4. False, since a student s eam grade is a function of te student s preparation. If two classmates ad te same IQ and onl one prepared ten te one wo prepared will most likel acieve a iger grade. 5. False, since ( + ) + + 6. False, since te domain is all real numbers. 7. True 8. True 9. True. Since an item as onl one price, b is a function of a. Since two items ma ave te same price, a is not a function of b.. a is not a function of b since tere ma be two students wit te same semester grades but different final eams scores. b is not a function of a since tere ma be identical final eam scores wit different semester grades.. a is not a function of b since it is possible tat two different students can obtain te same final eam score but te times spent on studing are different. b is not a function of a since it is possible tat 0. False, since 8, 8 and 8, 5 two different students can spend te same time are two ordered studing but obtain different final eam scores. pairs wit te same first coordinate and different second coordinates.. Eercises. relation. function. independent, dependent 4. domain, range So a is a function of b, and b is a function of a. 9. a is a function of b since a given denomination as a unique lengt. Since a dollar bill and a five-dollar bill ave te same lengt, ten b is not a function of a. 0. Since different U.S. coins ave different diameters, ten a is a function of b and b is a function of a. 5. difference quotient 6. average rate of cange 7. Note, b πa is equivalent to a b. π Ten a is a function of b, and b is a function of a. 8. Note, b (5 + a) is equivalent to a b 0.

4. a is not a function of b since it is possible tat two adult males can ave te same soe size but ave different ages. b is not a function of a since it is possible for two adults wit te same age to ave different soe sizes. 5. Since in.54 cm, a is a function of b and b is a function of a. 6. Since tere is onl one cost for mailing a first class letter, ten a is a function of b. Since two letters wit different weigts eac under /-ounce cost 4 cents to mail first class, b is not a function of a. 7. No 8. No 9. Yes 0. Yes. Yes. No. Yes 4. Yes 5. Not a function since 5 as two different second coordinates. 6. Yes 7. Not a function since as two different second coordinates. 8. Yes 9. Yes 0. Yes. Since te ordered pairs in te grap of 8 are (, 8), tere are no two ordered pairs wit te same first coordinate Coprigt 0 Pearson Education, Inc.

. FUNCTIONS 8 and different second coordinates. We ave a function.. Since te ordered pairs in te grap of + 7 are (, + 7), tere are no two ordered pairs wit te same first coordinate and different second coordinates. We ave a function.. Since ( + 9)/, te ordered pairs are (, ( + 9)/). Tus, tere are no two ordered pairs wit te same first coordinate and different second coordinates. We ave a function. 4. Since, te ordered pairs are (, ). Tus, tere are no two ordered pairs wit te same first coordinate and different second coordinates. We ave a function. 5. Since ±, te ordered pairs are (, ±). Tus, tere are two ordered pairs wit te same first coordinate and different second coordinates. We do not ave a function. 6. Since ± 9 +, te ordered pairs are 4. Since (, ) and (, ) are two ordered pairs wit te same first coordinate and different second coordinates, te equation does not define a function. 4. Since (, ) and (, ) are two ordered pairs wit te same first coordinate and different second coordinates, te equation does not define a function. 4. Domain {, 4, 5}, range {,, 6} 44. Domain {,,, 4}, range {, 4, 8, 6} 45. Domain (, ), range {4} 46. Domain {5}, range (, ) 47. Domain (, ); since 0, te range of + 5 is [5, ) 48. Domain (, ); since 0, te range of + 8 is [8, ) 49. Since, te domain of is [, ); range (, ) (, ± 9 + ). Tus, tere are two ordered 50. Since, te domain of pairs wit te same first coordinate and different second coordinates. We do not ave a function. is [, ); Since is a real number wenever 0, te range is [0, ). 7. Since, te ordered pairs are (, ). Tus, tere are no two ordered pairs wit te same first coordinate and different second coordinates. We ave a function. 8. Since, te ordered pairs are (, ). Tus, tere are no two ordered pairs wit te same first coordinate and different second coordinates. We ave a function. 9. Since, te ordered pairs are (, ). Tus, tere are no two ordered pairs wit te same first coordinate and different second coordinates. We ave a function. 40. Since +, te ordered pairs are (, + ). Tus, tere are no two ordered pairs wit te same first coordinate and different second coordinates. We ave a function. 5. Since 4 is a real number wenever 4, te domain of 4 is [4, ). Since 4 0 for 4, te range is [0, ). 5. Since 5 is a real number wenever 5, te domain of 5 is (, 5]. Since 5 0 for 5, te range is [0, ). 5. Since 0, te domain of is (, 0]; range is (, ); 54. Since 0, te domain of is (, 0]; range is (, ); 55. 6 56. 5 57. g() () + 5 58. g(4) (4) + 5 7

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8 CHAPTER FUNCTIONS AND GRAPHS 59. Since (, 8) is te ordered pair, one obtains Te average rate of cange on [.9, ] is f () 8. Te answer is. 60. Since (, 6) is te ordered pair, one obtains () (.9).9 0 6.4 0. 6.4 ft/sec. f () 6. Te answer is. Te average rate of cange on [.99, ] is 6. Solving + 5 6, we find 7. () (.99) 0 0.684.99 0.0 6.84 ft/sec. 6. Solving + 5 4, we find. Te average rate of cange on [.999, ] is 6. f (4) + g(4) 5 + 7 64. f () g() 8 4 6 () (.999).999 ft/sec. 0 0.06984 0.00 6.984 65. a a 66. w w 67. 4(a+) 4a+6 68. 4(a 5) 4a 69. ( + + ) ( + ) + 5 + 70. ( 6 + 9) ( ) 9 + 0 7. 4( + ) 4 + 4 7. ( ++ ) +6+ 7. ( + + ) ( + ) + 6 + 74. 4( + ) 4 + 8 80. 6 70 0 64 ft/sec 768 970 8. Te average rate of cange is 0 0. million ectares per ear. 8. If 0. million ectares are lost eac ear and 970 since 95 ears, ten te forest will 0. be eliminated in te ear 8 ( 988 + 95). 8. 75. ( + + ) ( + ) + f ( + ) f () 4( + ) 4 6 + 4 76. (4 + 4 ) 4 + 4 77. Te average rate of cange is 4 8, 000 0, 000 5 $, 400 per ear. 84. 78. Te average rate of cange as te number of cubic ards canges from to 0 and from 0 to 60 are 58 40 $6 per d and f ( + ) f () ( + ) 0 948 58 $4 per d, respectivel. 60 0 79. Te average rate of cange on [0, ] is

85. () (0) 0 64 ft/sec. f ( + ) f () ( + ) + 5 5 0 0 Te average rate of cange on [, ] is () () 0 48 48 ft/sec. Coprigt 0 Pearson Education, Inc.

. FUNCTIONS 8 86. f ( + ) f () ( + ) + + 9( + ) 9 ( + + ) 9 ( + + ) + + 87. Let g() +. Ten we obtain g( + ) g() ( + ) + ( + ) + + + +. 88. Let g(). Ten we get 9. Difference quotient is + + + + 4( + ) 4 ( + ) 4 ( + ) + + 9. Difference quotient is + + + + + + + g( + ) g() + + + + ( + + ) ( + ) ( + ) ( + ) + + +. ( + + + + ) ( + + + + ) + + + + 89. Difference quotient is 94. Difference quotient is r ( + ) + ( + ) + + + r + + r + + + r r + r + 90. Difference quotient is r r! + + ( + ) ( + ) + +

+ r r! + + + r r! + 9. Difference quotient is + + + + + + + + Coprigt 0 Pearson Education, Inc.

84 CHAPTER FUNCTIONS AND GRAPHS 95. Difference quotient is + ( + ) ( + ) ( + ) ( + ) 99. a) A s b) s A c) s d d) d s e) P 4s f ) s P /4 g) A P /6 ) d A r A 00. a) A πr b) r π d) d r e) d C π r f ) c) C πr A πd 4 ( + ) g) d A π ( + ) 0. C 50 + 5n 96. Difference quotient is 0. a) Wen d 00 ft, te atmosperic pressure + ( + ) is A(00).0(00) + 4 atm. ( + ) ( + ) ( + ) ( + ) 0. ( + ) 97. Difference quotient is + + + ( + + )( + ) ( + + )( + ) ( + ) ( + + ) ( + + )( + ) ( + + )( + ) 04. ( + + )( + ) 98. Difference quotient is + ( + )( ) b) Wen A 4.9 atm, te dept is found b solving 4.9 0.0d + ; te dept is.9 d 0.0 0 ft. (a) Te quantit C (4) (0.95)(4) + 5.8 $9.6 billion represents te amount spent on computers in te ear 004. (b) B solving 0.95n + 5.8 5, we obtain 9. n 0.95 0. Tus, spending for computers will be $5 billion in 00. (a) Te quantit E(4) + C (4) [0.5(4) + ] + 9.6 $.6 billion represents te total amount spent on electronics and computers in te ear 004.

( + )( ) ( ) ( + ) ( + )( ) ( + )( ) ( + )( ) (b) B solving (0.5n + ) + (0.95n + 5.8) 0.45n. n 9 we find tat te total spending will reac $0 billion in te ear 009 ( 000 + 9). Coprigt 0 Pearson Education, Inc.

. FUNCTIONS 85 (c) Te amount spent on computers is growing faster since te slope of C (n) [wic is ] is greater tan te slope of E(n) [wic is 0.95]. 05. Let a be te radius of eac circle. Note, triangle 4ABC is an equilateral triangle wit side a and eigt a. a d + 6d a ( 6 + )d a d 6 + 6 d a. 4 07. Wen 8 and 0., we ave A C B R(8.) R(8), 950. 0. Te revenue from te concert will increase b approimatel $,950 if te price of a ticket is raised from $8 to $9. If and 0., ten Tus, te eigt of te circle centered at C from te orizontal line is a + a. Hence, b using a similar reasoning, we obtain tat eigt of te igest circle from te line is a + a or equivalentl ( + )a. 06. In te triangle below, P S bisects te 90-angle at P and SQ bisects te 60-angle at Q. d S a R(.) R() 0., 050. Te revenue from te concert will decrease b approimatel $,050 if te price of a ticket is raised from $ to $. 08. Wen r.4 and 0., we obtain A(.5) A(.4) 0. 6. Te amount of tin needed decreases b approimatel 6. in. if te radius increases from.4 in. to.4 in. If r and 0., ten A(.) A() 0. 8.6 Te amount of tin needed increases b about 45 0 90 0 0 0 8.6 in. if te radius increases from in. to P R Q In te 45-45-90 triangle 4SP R, we find P R SR d/. And, in te 0-60-90 triangle 4SQR we get. in. 5 5 9 6 7 P Q 6 d. 8 8 Since P Q P R + RQ, we obtain 7

a d + 6 d 9 7 Coprigt 0 Pearson Education, Inc.

86 CHAPTER FUNCTIONS AND GRAPHS. If m is te number of males, ten m + m 6 m 6 m (6) 4 males p. ( 4 + 6) + ( ) 4 + 6 40 0. Pop Quiz. Yes, since A πr were A is te area of a circle wit radius r.. No, since te ordered pairs (, 4) and (, 4) ave te same first coordinates.. No, since te ordered pairs (, 0) and (, 0) ave te same first coordinates. 4. [, ) 5. [, ) 6. 9 4. Te slope is 5. 5 +. Te line is given 6 + 6 6. 6 6 4 0 ( 7)( + 6) 0 Te solution set is { 6, 7}. 6. Te inequalit is equivalent to 7. If a, ten a /. 40 0 b + b for some b. Substitute te 8. $ per ear 6 008 998 coordinates of (, ) as follows: 9. Te difference quotient is ( ) + b 6 f ( + ) f () ( + ) + b 6 + + Te line is given b +. Linking Concepts + (a) Te first grap sows U.S. federal debt D versus ear 6000 debt < 9 < 4 < < < < Te solution set is (, ). 000 500 00 ear 940 970 000 and te second grap sows population P (in millions) versus. population Tinking Outside te Bo XXII (0 + 5) 05 00 00 00 940 970 000 ear

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. GRAPHS OF RELATIONS AND FUNCTIONS 87 (b) Te first table sows te average rates of cange for te U.S. federal debt 0 ear period ave. rate of cange 57 5 940 50 0 0.6 9 57 950 60 0.4 8 9 960 70 0 9.0 909 8 970 80 0 5.8 07 909 980 90 0 9.8 990 000 45.9 5666 07 0 Te second table sows te average rates of cange for te U.S. population 0 ear period ave. rate of cange 50.7.7 940 50 0.9 79. 50.7 950 60 0.9 0. 79. 960 70 0.4 6.5 0. 970 80 0. 48.7 6.5 980 90 0. 74.8 48.7 990 000 0.6 (c) Te first table sows te difference between consecutive average rates of cange for te U.S. federal debt. 0-ear periods difference 940-50 & 950-60.4 0.6 7. 950-60 & 960-70 9.0.4 5.6 960-70 & 970-80 5.8 9.0 4.8 970-80 & 980-90 9.8 5.8 77.0 980-90 & 990-00 45.9 9.8 6. Te second table sows te difference between consecutive average rates of cange for te U.S. population. 0-ear periods difference 940-50 & 950-60.9.9.0 950-60 & 960-70.4.9 0.5 960-70 & 970-80..4 0. 970-80 & 980-90.. 0. 980-90 & 990-00.6. 0.4 (f ) Te U.S. federal debt is growing out of control wen compared to te U.S. population. See part (g) for an eplanation. (g) Since most of te differences for te federal debt in part (e) are positive, te federal debts are increasing at an increasing rate. Wile te U.S. population is increasing at a decreasing rate since most of te differences for population in part (e) are negative. For Tougt. True, since te grap is a parabola opening down wit verte at te origin.. False, te grap is decreasing.. True 4. True, since f ( 4.5) [.5]. 5. False, since te range is {±}. 6. True 7. True 8. True 9. False, since te range is te interval [0, 4]. 0. True. Eercises. parabola. piecewise. Function includes te points (0, 0), (, ), domain and range are bot (, ) 4 (d) For bot te U.S. federal debt and population, te average rates of cange are all positive. (e) In part (c), for te federal debt most of te differences are positive and for te population most of te differences are negative. Coprigt 0 Pearson Education, Inc.

88 CHAPTER FUNCTIONS AND GRAPHS 4. Function includes te points (0, 0), (, ), (, ), domain and range are bot (, ) 9. Function includes te points (0, 0), (±, ), domain is (, ), range is [0, ) 8-5. Function 0 includes te points (, ), (0, 0), (, ), domain and range are bot (, ) 0. Function goes troug (0, ), (±, 0), domain is (, ), range is [, ) 4-6. Function includes te points (, 0), (0, ), (, 4), domain and range are bot (, ). Function includes te points (0, ), (±, 0), domain is (, ), range is (, ] - -4 7. Function 5 includes te points (0, 5), (±, 5), domain is (, ), range is {5} -4. Function includes te points (0, ), (±, ), domain is (, ), range is (, ] 6 4-5 5-4 8. is not a function and includes te points (, 0), (, ), domain is {}, range is (, ) 4 - Coprigt 0 Pearson Education, Inc.

. GRAPHS OF RELATIONS AND FUNCTIONS 89. Function + includes te points (0, ), (, ), (4, ), domain is [0, ), range is [, ) 7. Function goes troug (0, 0), (, 4), (, 9), domain and range is [0, ) 9 4 4 4 4. Function includes te points (0, ), 8. Function goes troug (, 0), (4, 0), domain is [0, ), range is (, ] (, 4), (4, 9), domain [, ), and range [0, ), 4-4 -4 5. + is not a function and includes te points (, 0), (, ±), domain is [, ), range is (, ) 9. Function + goes troug (, 0), (, ), (8, ), domain (, ), and range (, ) - - 4 4 6. is not function and includes te points (, 0), (0, ±), domain is (, ], range is (, ) - 0. Function goes troug (, ), (, ), (8, 0), domain (, ), and range (, ) - 4 - - Coprigt 0 Pearson Education, Inc.

90 CHAPTER FUNCTIONS AND GRAPHS. Function, goes troug (0, 0), (, ), (, 8), domain (, ), and range (, ) 4. Not a function, + 4 goes troug (, 0), (0, ), (, 0), domain [, ], and range [, ] - 8 8. Function, goes troug (0, ), (, ), (, 0), domain (, ), and range (, ) 5. Function, goes troug (±, 0), (0, ), domain [, ], and range [0, ] - - -. Not a function, goes troug 6. Function, 5 goes troug (, 0), (0, ), (, 0), domain [, ], and range [, ] (±5, 0), (0, 5), domain [ 5, 5], and range [ 5, 0] - 4 6 - -4-6

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. GRAPHS OF RELATIONS AND FUNCTIONS 9 7. Function includes te points (0, 0), (, ), (, 8), domain and range are bot (, ). Function includes te points (0, 0), (±, ), domain is (, ), range is (, 0] 8 - -4 8. Function includes te points (0, 0), (, ), (, 8), domain and range are bot (, ). Function + includes te points (, 0), (0, ), (, ), domain is (, ), range is (, 0] 8 4 - - 4-4 -8 9. Function includes te points (0, 0), (±, ), domain is (, ), range is [0, ) 5. Not a function, grap of includes te points (0, 0), (, ), (, ), domain is [0, ), range is (, ) 0. Function includes te points (0, ), (, 0), (, ), domain is (, ), range is [0, ) 6-4. + is not a function and includes te points (, 0), (, ±), domain is [, ), range is (, ) - Coprigt 0 Pearson Education, Inc.

9 CHAPTER FUNCTIONS AND GRAPHS 5. Domain is (, ), range is {±}, some points are (, ), (, ) 9. Domain is [, ), range is (, ], some points are (, ), (, 0), (, ) - - 6. Domain is (, ), range is {, }, some points are (0, ), (4, ) - 40. Domain is (, ), range is (, ), some points are (, ), (4, ), (, ) 4 - - - 4 7. Domain is (, ), range is 4. Domain is (, ), range is [0, ), some (, ] (, ), some points are (, ), points are ( 4, ), (4, ), ), ( (, ) 4-8. Domain is (, ), range is [, ), some points are (, ), (, 4) 4. Domain is (, ), range is [, ), some points are (, ), (0, ), (, 4) 4 - Coprigt 0 Pearson Education, Inc.

. GRAPHS OF RELATIONS AND FUNCTIONS 9 4. Domain is (, ), range is (, ), some points are (, 4), (, ) 47. Domain [0, 4), range is {,, 4, 5}, some points are (0, ), (, ), (.5, ) 5-44. Domain is [, ), range is [0, ), some points are (±, 0), (, ) 5-48. Domain is (0, 5], range is {,,, 0,, }, some points are (0, ), (, ), (.5, ) 5-49. a. Domain and range are bot (, ), decreasing on (, ) b. Domain is (, ), range is (, 4] 45. Domain is (, ), range is te set of inte- increasing on (, 0), decreasing on (0, ) - 46. Domain is (, ), range is te set of even integers, some points are (0, 0), (, ), (.5, ) increasing on (, ) b. Domain is (, ), range is [, ) increasing on (0, ), decreasing on (, 0) 5. a. Domain is [, 6], range is [, 7] increasing on (, ), decreasing on (, 6) b. Domain (, ], range (, ], increasing on (, ), constant on (, ) 5. a. Domain is [0, 6], range is [ 4, ] - - b. Domain (, ), range [, ), decreasing on (, ) 5. a. Domain is (, ), range is [0, ) increasing on (0, ), decreasing on (, 0) b. Domain and range are bot (, ) increasing on (, /), decreasing on (, ) and ( /, ) Coprigt 0 Pearson Education, Inc.

94 CHAPTER FUNCTIONS AND GRAPHS 54. a. Domain is [ 4, 4], range is [0, 4] increasing on ( 4, 0), decreasing on (0, 4) b. Domain is (, ), range is [, ) increasing on (, ), decreasing on (, ), constant on (, ) 55. a. Domain and range are bot (, ), increasing on (, ) b. Domain is [, 5], range is [, 4] increasing on (, ), decreasing on (, ), constant on (, 5) 56. a. Domain is (, ), range is (, ] increasing on (, ), decreasing on (, ) b. Domain and range are bot (, ), decreasing on (, ) 57. Domain and range are bot (, ) increasing on (, ), some points are (0, ), (, ) 60. Domain is (, ), range is [, ), increasing on (0, ), decreasing on (, 0), some points are (0, ), (, ) 6. Domain is (, 0) (0, ), range is {±}, constant on (, 0) and (0, ), some points are (, ), (, ) - - 5 58. Domain and range are bot (, ), decreasing on (, ), some points are (0, 0), (, ) 6. Domain is (, 0) (0, ), range is {±}, constant on (, 0) and (0, ), some points are (, ), (, ) - 6 6. Domain is [, ], range is [0, ], increasing on (, 0), decreasing on (0, ), -6 some points are (±, 0), (0, ) 4 59. Domain is (, ), range is [0, ), increasing on (, ), decreasing on (, ), some points are (0, ), (, 0) 4 5 Coprigt 0 Pearson Education, Inc.