Eercises. CHAPTER Functions EXERCISES.. { 0 < 6}. a. Since and m, ten y, te cange in y, is y m. { } 7. For (, ) and (, ), te slope is Since and m, ten y, te cange in y, is y m 0 9. For (, 0) and (, ), te slope is 0 ( ) + 6. For (, ) and (, ), te slope is ( ) + undefined 0. For (0, ) and (, ), te slope is + 0 0 0. Since y is in slope-intercept form, m and te y-intercept is (0, ). Using te slope m, we see tat te point unit to te rigt and units up is also on te line. 7. Since y is in slope-intercept form, m and te y-intercept is (0, 0). Using m, we see tat te point units to te rigt and unit down is also on te line. 9. Te equation y is te equation of te orizontal line troug all points wit y-coordinate. Tus, m 0 and te y-intercept is (0, ). Copyrigt Hougton Mifflin Company. All rigts reserved.
Capter : Functions. Te equation is te equation of te vertical line troug all points wit -coordinate. Tus, m is not defined and tere is no y-intercept.. First, solve for y: y y + y Terefore, m and te y-intercept is (0, ).. First, solve for y: + y 0 y Terefore, m and te y-intercept is (0, 0). 7. First, solve for y: y 0 y y Terefore, m and te y-intercept is (0, 0). 9. First, put te equation in slope-intercept form: y + y + Terefore, m and te y-intercept is 0,.. First, solve for y: y y + y Terefore, m and te y-intercept is (0, )... 7. on [ 0, 0] by [ 0, 0] on [ 0, 0] by [ 0, 0] 9. y. + on [ 7, 7] by [ 7, 7] Copyrigt Hougton Mifflin Company. All rigts reserved.
Eercises. [ ]. y y + +. y y +.. 7. First, find te slope. m 7 Ten use te point-slope formula wit tis slope and te point (, ). y ( ) y +0 y + 9. First, find te slope. m + 0 Ten use te point-slope formula wit tis slope and te point (, ). y 0( ) y + 0 y. Te y-intercept of te line is (0, ), and y for. Tus, m y. Now, use te slope-intercept form of te line: y +.. Te y-intercept is (0, ), and y for. Tus, m y. Now, use te slopeintercept form of te line: y. First, consider te line troug te points (0, ) and (, 0). Te slope of tis line is m 0 0. Since (0, ) is te y-intercept of tis line, use te slope-intercept form of te line: y + or y +. Now consider te line troug te points (, 0) and (0, ). Te slope of tis line is m 0 0. Since (0, ) is te y-intercept of te line, use te slope-intercept form of te line: y or y Net, consider te line troug te points (0, ) and (, 0). Te slope of tis line is m 0 ( ). Since (0, ) is te y-intercept, use te slope-intercept form of te line: y or y Finally, consider te line troug te points (, 0) and (0, ). Te slope of tis line is m (0, ) is te y-intercept, use te slope-intercept form of te line: y + or y + 0 0 0. Since 7. If te point (, y ) is te y-intercept (0, b), ten substituting into te point-slope form of te line gives y y m y b m( 0) y b m y m+ b 9. a. on [, ] by [, ] on [, ] by [, ] Copyrigt Hougton Mifflin Company. All rigts reserved.
Capter : Functions 6. Low demand: [0, 8); average demand: [8, 0); ig demand: [0, 0); critical demand: [0, ) 6. a. Te value of corresponding to te year 00 is 00 900 0. Substituting 0 into te equation for te regression line gives y 0.6+ 7. y 0.6(0) + 7. 8.8 seconds Since minutes 80 seconds, 8. minutes 8.8 seconds. Tus, te world record in te year 00 will be minutes 8.8 seconds. To find te year wen te record will be minutes 0 seconds, first convert minutes 0 seconds to seconds: minutes 0 seconds minutes 60 sec min + 0 seconds 0 seconds. Now substitute y 0 seconds into te equation for te regression line and solve for. y 0.6+ 7. 0 0.6 + 7. 0.6 7. 0 0.6 7. 7..6 0.6 Since represents te number of years after 900, te year corresponding to tis value of is 900 +.6 0.6 0. Te world record will be minutes 0 seconds in 0. 6. a. To find te linear equation, first find te slope of te line containing tese points. m 6 8 Net, use te point-slope form wit te point (, 6): y y m ( ) y 6 ( ) y 6 y + To find te profit at te end of years, substitute into te equation y +. y () + $0 million c. Te profit at te end of years is y () + $ million. 69. a. Price $0,000; useful lifetime 0 years; scrap value $6000 V 0,000 0,0006000 t 0 t 0 0 0,000 00t 0 t 0 67. a. First, find te slope of te line containing te points. m 80 9 00 0 00 Net, use te point-slope form wit te point (0, ): y y m y 9 ( 0) y 9 + Substitute 0 into te equation. y 9 + y 9 (0) + 6 + 68 F 7. a. Median Marriage Age for Men on [0, ] by [0, ] Median Marriage Age for Women on [0, ] by [0, ] Copyrigt Hougton Mifflin Company. All rigts reserved.
Eercises. 69. Substitute t into te equation. V 0,000 00t 0,000 00() 0,000,000 $9, 000 7. Te -value corresponding to te year 00 is 00 980 0. Te following screens are a result of te EVALUATE command wit 0. c. Median Age at Marriage Median Age at Marriage for Men in 00 for Women in 00. So, te median marriage age for men in 00 will be 8.7 years and for women it will be 7. years. c. Te -value corresponding to te year 00 is 00 980 0. Te following screens are a result of te EVALUATE command wit 0. on [0, 0] by [0, 0,000] Median Age at Marriage Median Age at Marriage for Men in 00 for Women in 00. So, te median marriage age for men in 00 will be 0 years and for women it will be 8.9 years. 7. a. 7. a-b on [0, 00] by [0, 0] To find te probability tat a person wit a family income of $0,000 is a smoker, substitute 0 into te equation y 0. + 0 y 0.(0) + 0 7.6 or 8%. c. Te probability tat a person wit a family income of $70,000 is a smoker is y 0.(70) + 0 8. or 8%. On [ 0, 6] by [6, 8] y 0.0 + 6.9. c. Te -value corresponding to te year 0 is 0 960 6. Substitute 6 into te equation y 0.0+ 6.9. y 0.0+ 6.9 y 0.0(6) + 6.9 79. years So, te life epectancy of a male cild born in te year 0 is approimately 79. years. 77. False: Infinity is not a number. 79. y y m for any two points (, y ) and (, y ) on te line or te slope is te amount tat te line rises wen increases by. 8. False: Te slope of a vertical line is undefined. 8. True: Te slope is a b and te y-intercept is c b. Copyrigt Hougton Mifflin Company. All rigts reserved.
6 Capter : Functions 8. To find te -intercept, substitute y 0 into te equation and solve for : 0 y m m + + b b m b b m If m 0, ten a single -intercept eists. So a b. Tus, te -intercept is m ( b,0 m ). 87. Consider R > and 0 < < K < K means tat K > 0 and 0< <. K Since K > 0, ten K + R> R K + ( R ) > R K + R > R K Terefore, K > R y + R K Additionally, since 0< <, K + ( R ) < + ( R) K So, y R > R R + R ( R ) R + K We ave < y < K. EXERCISES. 6 6. 6.. 8 9. () 8 8 7.. 9 8 7. /. 8 7. 9. /. / ( ) 6 6 8. ( ). ( ) / 8 8 / / 8 8 7. 7 7 9 /. 6 6 6 6 6 / 9... / 7. 8. ( 8) 8 9. ( 8) ( 8) ( 8) / 8 8 8 8. 6 6 6 9 Copyrigt Hougton Mifflin Company. All rigts reserved.
Eercises. 7. 6 6 6 6 9. 0.9 7...7 8 7.7. 7 8 8 7.. 0. 6. 0 0. 0. 0. 0.977 7. 7 7 7 6.. 9. 7 7 0. 0. 6. 000.70 0 6. ( ) ( ) 6 67. z ( z z ) z z ( z ) z ( z z z) 9 7 ( z ) z 69. 7. 8 7. 000 9 ( ww ) w w w ww w w ( ) 8 7. 9y z 6 8 y z y y y y y ( uvw) 77. ( uw ) 8. C 0.6 C 6 uvw uvw uw 0.6 C.C To quadruple te capacity costs about. times as muc. ( yz) 7y y z 79. Average body tickness 8. 0.(ip-to-soulder lengt) / 0.(6) 0. 6.6 ft on [0, ] by [0, ] / Capacity can be multiplied by about. 8. ( Heart rate) 0( weigt) / 0( 6) / beats per minute 87. on [0, 00] by [0, 0] Heart rate decreases more slowly as body weigt increases. Copyrigt Hougton Mifflin Company. All rigts reserved.
8 Capter : Functions 89. (Time to build te 0t Boeing 707) 0. 0(0).6 tousand work-ours It took approimately,600 work-ours to build te 0t Boeing 707. 9. S 60 0. 60 ( 8)0. mp 9. a. Increase in ground motion 0 B A 9. 0 8.6.8 0. Te 906 San Francisco eartquake ad about times more grand motion tan te 99 Nortridge eartquake. Increase in ground motion 0 0 BA 9.07..8 0 6 Te 00 Sumatra eartquake ad about 6 times more ground motion tan te 99 Kobe eartquake. on [0,00] by [0,] 8.. Terefore, te land area must be increased by a factor of more tan 8 to double te number of species. 97. a. 99., since 9 means te principal square foot. (To get ± you would ave to write ± 9.) 0. on [, ] by [000, 00] y 6 0.67 c. For 0, 0.67 y 6 0.67 6( 0) 7 work-ours It will take approimately 7 workours to build te fiftiet supercomputer. False: 6 6 6 6, wile 6/ 8. m mn.) n (Te correct statement is 0. /, so must be nonnegative for te epression to be defined. 0., so all values of ecept 0, because you cannot divide by 0. Copyrigt Hougton Mifflin Company. All rigts reserved.
Eercises. 9 EXERCISES.. Yes. No. No 7. No 9. Domain { 0 or } Range {y y }. a. f f (0) 0 9. a. Domain { } since f is defined for all values of.. a. z z + ( ) + Domain {z z } since ( z) z+ is defined for all values of z ecept z. c. Range {y y 0} c. Range {y y 0} / / (8) 8 8 7. a. f / / f ( 8) ( 8) 8 / Domain { 0} since is Domain defined only for nonnegative values of. c. Range {y y 0} c. Range {y y 0} 9. a. f f (0) 0 f ( ) is defined for values of suc tat 0. Tus, 0 Domain { } c. Range {y 0 y }. a. f f ( ) ( ) f is defined only for values of suc tat > 0. Tus 0. Domain { 0} c. Range {y y 0}.. 7. 9. Copyrigt Hougton Mifflin Company. All rigts reserved.
0 Capter : Functions. a. a b 0 () 0 0 To find te y-coordinate, evaluate f 0. f ( 0) ( 0) 0( 0)+ 00 00 Te verte is (0, 00). at. a. b ( 80) 80 0 a To find te y-coordinate, evaluate f 0. f ( 0) ( 0) 80( 0) 800 00 Te verte is ( 0, 00). at on [, ] by [00, 0] on [, ] by [ 0, 00]. 6 7 0 ( 7) ( + ) 0 at 7 at 7, 9. + 0 8 8 + 0 0 9 + 0 0 ( ) ( ) 0 at at,. 0 0 0 ( ) ( + ) 0 at at, 7. + 8 + 8 0 + 0 ( ) ( ) 0 at at,. + 0 + 0 ± Undefined + 0 as no real solutions. 7. + + 0 ( + ) ( ) 0 at at,. 0 0 0 0 ( 0) 0. at 0 at 0 0, 0 + + 0 + + 6 0 + 6+ 9 0 ( + ) 0 Equals 0 at 9. + 0 0 6 + 0 0 Use te quadratic formula wit a, b 6, and c 0. 6 ± ( 6 ) ()0 () 6± 6 0 6± Undefined + 0 0 as no real solutions.. on [, 6] by [, 6], Copyrigt Hougton Mifflin Company. All rigts reserved.
Eercises.. 7. 9. on [, 9] by [ 0, 0], 6. on [ 7, ] by [, 6] 6. on [, ] by [, 0] No real solutions on [, ] by [ 9, ].6,. 6. Let te number of board feet of wood. Ten C() + 0 on [ 0, 0] by [ 0, 0] a. Teir slopes are all, but tey ave different y-intercepts. Te line units below te line of te equation y 6 must ave y-intercept 8. Tus, te equation of tis line is y 8. 67. Let te number of ours of overtime. Ten P() + 00 7. Dv () 0. 0v +.v D( 0) 0. 0( 0) +.( 0) ft 7. v () 60 v( ) 60 08 mp 69. a. pd 0. d + p() 6 0. ()+ 6 7. 7 pounds per square inc pd () 0.d + p(, 000) 0.(, 000)+,76 pounds per square inc 7. a. 77. Nt () 00+ 0t N() 00+ 0() 00 cells Nt () 00+ 0t N(0) 00+ 0(0) 00 cells on [0, ] by [0, 0] Te object its te ground in about.9 seconds. Copyrigt Hougton Mifflin Company. All rigts reserved.
Capter : Functions 79. a. To find te break-even points, set C() equal to R() and solve te resulting equation. C R 80+ 6, 000 + 660 80+ 6,000 0 Use te quadratic formula wit a, b 80 and c 6,000. 80 ± ( 80) ()(6, 000) () 80± 0,00 80± 0 800 or 60 00 or 0 Te company will break even wen it makes eiter 0 devices or 00 devices. To find te number of devices tat maimizes profit, first find te profit function, P() R() C(). P ( + 660 ) (80+ 6,000) + 80 6, 000 Since tis is a parabola tat opens downward, te maimum profit is found at te verte. 80 80 0 Tus, profit is maimized wen 0 devices are produced per week. Te maimum profit is found by evaluating P(0). P (0) (0) + 80(0) 6, 000 $,800 Terefore, te maimum profit is $,800. 8. a. To find te break-even points, set C() equal to R() and solve te resulting equation. C R () 00 + 00 + 00 00 + 00 0 Use te quadratic formula wit a, b 00 and c 00. 00 ± ( 00) ()(00) () 00±,00 00± 0 0 or 80 80 or 0 Te store will break even wen it sells eiter 0 eercise macines or 80 eercise macines. To find te number of eercise macines tat maimizes profit, first find te profit function, P() R() C(). P ( + 00 ) (00+ 00) + 0000 Since tis is a parabola tat opens downward, te maimum profit is found at te verte. 00 00 0 Tus, profit is maimized wen 0 eercise macines are sold per day. Te maimum profit is found by evaluating P(0). P( 0) 0 + 00( 0) 00 $800 Terefore, te maimum profit is $800. 8. w+ a v+ b c v+ b c w+ a v c b w+ a Copyrigt Hougton Mifflin Company. All rigts reserved.
Eercises. 8. a. 87. a. (00 ) 00 or + 00 f 00 or f + 00 On [0, 6] by [0, 00]. y 0.8 8.+ 7. y 0.8() 8.() + 7. y 9.76 Te probability tat a ig scool graduate smoker will quit is 0%. c. y 0.8 8.+ 7. y 0.8(6) 8.(6) + 7. y 60.6 Te probability tat a college graduate smoker will quit is 60%. Since tis function represents a parabola opening downward (because a ), it is maimized at its verte, wic is found using te verte formula, b, wit a and b 00. a 00 0 Se sould carge $0 to maimize er revenue. 88. a. Te upper limit is f 0.7(0 ) 0.7 Te lower limit is f 0.(0 ) 0 0. Te lower cardio limit for a 0-year old is g (0) 0 0.(0) 00 bpm Te upper cardio limit for a 0-year old is (0) 0.7(0) 0 bpm f Te lower cardio limit for a 60-year old is g (60) 0 0.(60) 80 bpm Te upper cardio limit for a 60-year old is (60) 0.7(60) bpm f 89. A function can ave more tan one -intercept. Many parabolas cross te -ais twice. A function cannot ave more tan one y-intercept because tat would violate te vertical line test. 90. f () 9, (since te two given values sow tat increasing by means y increases by.). 9. Because te function is linear and is alfway between and 6, f () 9 (alfway between 7 and ). 9. m is blargs per prendle and prendles and y is in blargs. y, so is in 9. No, tat would violate te vertical line test. Note: A parabola is a geometric sape and so may open sideways, but a quadratic function, being a function, must pass te vertical line test. Copyrigt Hougton Mifflin Company. All rigts reserved.
Capter : Functions EXERCISES.. Domain { < or >0} Range {y y < or y > 0}. a. f + f ( ) +. a. f f Domain { } Domain { } c. Range {y y 0} c. Range {y y 0 or y } 7. a. f() ( +) f () + 9. a. g + g( ) ( )+ Domain { 0, } Domain c. Range {y y or y > 0} c. Range {y y 0}. + 0 ( + ) 0 ( + ) ( ) 0 Equals 0 at 0 at at 0,,and. +8 +8 0 ( 6 + 9) 0 ( ) 0 at 0 at 0 and. 0 0 ( ) 0 ( ) ( + ) 0 Equals 0 at 0 at at 0,,and 7. 6 0 6 0 0 6 ( ) 0 at 0 at 0 and 9. / / / 6 9 / / / 6 9 0 / ( ) 0 / ( )( + ) 0 Equals 0 at 0 at at 0, and. on [, ] by [ 0, 0], 0, and Valid solutions are 0 and... on [, ] by [ 0, ], 0, and on [, ] by [ 0, 0] 0 and Copyrigt Hougton Mifflin Company. All rigts reserved.
Eercises. 7. 9. on [ 6, ] by [ 00, 600] and 0 on [0, ] by [,] 0 and.. on [, ] by [, 0].79, 0, and.79. 7. y 9.. Polynomial on [, 0] by [, ]. Piecewise linear function. Polynomial 7. Rational function 9. Piecewise linear function. Polynomial. None of tese. a. 7. a. f( g() ) [ g ()] ( 7 ) g( f( ) 7[ f() ] 7( ) 7 on [, ] by [, ] Note tat eac line segment in tis grap includes its left end-point, but ecludes its rigt endpoint. So it sould be drawn like. Domain ; range te set of integers 9. a. f( g() ) g () + g( f() ) f() 9. 6. a. f( g( ) g () [ ] [ g ] ( ) [ ] + + g( f() ) f () Copyrigt Hougton Mifflin Company. All rigts reserved.
6 Capter : Functions 6. a. f( g() [ ) g ()] g [()] + g ( f ) f f + + ( ) ( ) + 67. ( f + + + + ) + 0+ 7. f f( + ) f ( + ) ( + + ) + 0 + 0 + (0+ ) 0 + or ( + ) 7. f 7 + f( + ) f 7( + ) ( + ) + (7 + ) 7( + + ) ( + ) + (7 + ) 7 + + 7 + 7 + + 7 (+ 7) + 7 6. a. f ( g ) g ( ) g f 6 + 6 + 66 f + 6+ + 69. f ( + ) ( + ) ( + ) + + + + 7. f + f( + ) f ( + ) ( + ) + ( + ) ( + + ) ( + ) + ( + ) + + + + + (+ ) + 77. f f( + ) f ( + ) + + + + + ( + + ) + + Copyrigt Hougton Mifflin Company. All rigts reserved.
Eercises. 7 79. f f( + ) f + ( + ) + ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) 8. f f( + ) f ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( ) ( + ) or ( + ) ( + + ) or + + 8. a..708.78 c..788 d. Yes,.788 8. Te grap of y ( + ) + 6 is te same sape as te grap of y but it is sifted left units and up 6 units. Ceck: on [ 0, 0] by [ 0, 0] 87. P (.00) P( 0) (.00) 0 680 million people in 70 89. a. For 000, use ƒ() 0.0. f 0.0 f( 000) 0.0( 000) $00 For 000, use ƒ() 0.0. f 0.0 f 000 0.0(000) $00 c. For 0,000, use ƒ() 00 + 0.0( 000). f 00 + 0.0( 000) f ( 0, 000) 00 + 0.0( 0, 000 000) 00 + 0.0( 000) $000 d. Copyrigt Hougton Mifflin Company. All rigts reserved.
8 Capter : Functions 9. a. For, use f 0.. f 0. f ( 0. 7 years For, use f 0.. f 0. f ( 0. years For, use f +. f + f () + ( ) + () 9 years For 0, use f +. f + f (0) + (0 ) + (8) years 9. First find te composition R(v(t)). ( ()) v() t 0. 0. R v t (60 + t) Ten find R(v(0)). ( ( 0) ) ( 60 (0) ) ( 90) R v + $7.7 million 0. 0. 9. a. 0 f( ), 08, 76 cells million cells 97. a. on [.6, 0] by [0, 000] f (),07, 7,8 cells At about 7.9 mpg No, te mouse will not survive beyond day. 99. One will ave missing points at te ecluded -values. 0. ( ) ( ) f f f + a ( + a) + a + a 0. f ( + 0) is sifted to te left by 0 units. 0. False: f( + ) ( + ) + +, not +. 07. a. y, because it is te polynomial function wit te largest degree. y 00, because it is te polynomial function wit te smallest degree. 09. a. ( ) f g a g + b a c+ d + b ac + ad + b Yes Copyrigt Hougton Mifflin Company. All rigts reserved.
Review Eercises 9 REVIEW EXERCISES. { < }. { < 0}. { 00}. { 6}. Hurricane: [7, ); storm: [, 7); gale: [8, ); small craft warning: [,8) 7. y y + y 9. Since te vertical line passes troug te point wit -coordinate, te equation of te line is.. First, calculate te slope from te two points. m 6 Now use te point-slope formula wit tis slope and te point (, ). y y y +. Since te y-intercept is (0, ), b. To find te slope, use te slope formula wit te points (0, ) and (, ). m 0 Tus te equation of te line is y.. a. Use te straigt-line depreciation formula wit price,000, useful lifetime 8, and scrap value 000. price scrap value Value price t useful lifetime, 000,000000 t 8, 000,000 8 t, 000 000t Value after years, 000 000( ), 000,000 $, 000 6. a. c. d. (0, ) (, 0) [0, ) (, 0] [ ] 8. y 6 y 6 y + 0. Since te orizontal line passes troug te point wit y-coordinate, te equation of te line is y.. First, calculate te slope from te points. m 6 Now, use te point-slope formula wit tis slope and te point (, ). y ( ) y+ y. Since te y-intercept is (0, ), b. To find te slope, use te slope formula wit te points (0, ) and (, 0). m 0 0 Te equation of te line is y + 6. a. Use te straigt-line depreciation formula wit price 78,000, useful lifetime, and scrap value 000. price scrap value Value price t useful lifetime 78,000 78,000000 t 78,000 7,000 t 78,000 000t Value after 8 years 78, 000 000( 8) 78, 000 0, 000 $8, 000 Copyrigt Hougton Mifflin Company. All rigts reserved.
0 Capter : Functions 7. a. 8. a. on [0, 0] by [0, 0] on [0, 0] by [0, 00] on [0, 0] by [0, 0] Te regression line y 0. +. fits te data reasonably well. on [0, 0] by [0, 00] Te regression line y. +. fits te data reasonably well. 9. 6 6 6 0. ( ). 6 6 8.. 000 000 0 ( 8) 8 8 7 / /.. 00 00 00 0 000 6. / / 8 7 7 9 6 6 7 8 8 6 6 9 9 7 9 7..97 8.. 9. a. 0.7 y 0.86 0.7 y 0.86(000). Te weigt for te top cold-blooded meat-eating animals in Hawaii is. lbs. 0.7 y 0.86 0.7 y 0.86(9, 00, 000) 68.8 Te weigt for te top cold-blooded meat-eating animals in Nort America is 68.8 lbs. 0. a. 0. y.7 0. y.7(000) 6.9 Te weigt for te top warm-blooded plant-eating animals in Hawaii is 6.9 lbs. 0. y.7 0. y.7(9, 00, 000) 78.8 Te weigt for te top warm-blooded plant-eating animals in Hawaii is 78.8 lbs.. a. f 7. a. Domain { 7} because 7 is defined only for all values of 7. g + Domain {t t } c. Range {y y 0} c. Range {y y 0} Copyrigt Hougton Mifflin Company. All rigts reserved.
Review Eercises / / 6 6. a. ( 6) 6 8 Domain {w w > 0} because te fourt root is defined only for nonnegative numbers and division by 0 is not defined. / / 8 8. a. w( 8) 8 6 Domain {z z 0} because division by 0 is not defined. c. Range {y y > 0} c. Range {y y > 0}. Yes 6. No 7. 8. 9. 0.. a. + 9 0 ( + ) 0 at 0 at 0 and Use te quadratic formula wit a, b 9, and c 0 9± 9 0 9± 8 6 9± 9 6 0, 0 and. a. 8 0 0 ( ) 0 ( )( + ) 0 at at and Use te quadratic formula wit a, b 8, and c 0 ( 8) ± ( 8) ()( 0) () and 8± 6+ 80 8±, Copyrigt Hougton Mifflin Company. All rigts reserved.
Capter : Functions. a. + + + 6 0 ( + ) 0 ( + )( ) 0 at at and ± ( 6) ± 9+ 7 6 ± 8 6 ± 9 6, and. a. Use te verte formula wit a and b 0. b a 0 () To find y, evaluate f(). f () 0 0 Te verte is (, 0).. a. ± and 0± 0 ± 6 8 and ± 8 8 ± 6. a. Use te verte formula wit a and b. b a () 7 To find y, evaluate f( 7). f( 7) ( 7) + ( 7) 6 Te verte is ( 7, 6) on [, ] by [ 0, 0] 7. Let number of miles per day. C() 0. + 9. Let te altitude in feet. T 70 00 on [ 0, 0] by [ 6, 6] 8. Use te interest formula wit P 0,000 and r 0.08. I(t) 0,000(0.08)t 800t 0. Let t te number of years after 000. C t t+ 0.8 7 0 0.8t + 7 t years after 000; in te year 00 Copyrigt Hougton Mifflin Company. All rigts reserved.
Review Eercises. a. To find te break even points, solve te equation C() R() for. C () R () 80 + 90 + 0 60 + 90 0 80 + 97 0 ( 6) ( ) 0 at 6 at 6 and Te store breaks even at receivers and at 6 receivers. To find te number of receivers tat maimizes profit, first find te profit function, P() R() C(). ( P + 0) ( 80+ 90) + 6090 Since tis is a parabola tat opens downward, te maimum profit is found at te verte. b 60 60 0 a Tus, profit is maimized wen 0 receivers are installed per week. Te maimum profit is found by evaluating P(0). P( 0)0 +60( 0)90 $0 Terefore, te maimum profit is $0.. a. To find te break even points, solve te equation C() R() for. C R 0 + 0, 00 + 00 800+ 0, 00 0 600+ 67, 00 0 0 0 0 at 0 at 0 0 and 0 Te outlet breaks even at 0 units and 0 units. To find te number of units tat maimizes profit, first find te profit function, P() R() C(). ( + 00) ( 0+ 0, 00) P + 8000, 00 Since tis is a parabola tat opens downward, te maimum profit is found at te verte. b 800 800 00 a 6 Tus, profit is maimized wen 00 units are installed per mont. Te maimum profit is found by evaluating P(00). P 00 00 + 800 00 0, 00 $67,00 Terefore, te maimum profit is $67,00.. a. f ( ). a. f ( 8) 6 6 ( 8)( 8+ ) Domain { 0, } Domain { 0, } c. Range {y y > 0 or y } c. Range {y y > 0 or y }. a. g ( ) + 0 6. a. Domain R Domain R g 0 c. Range {y y } c. Range {y y 0} 7. + 0 + 0 0 + 0 + 0 Equals 0 at 0 at at 0,, and 8. + 8 + 8 0 + 8 0 + 0 Equals 0 at 0 at at 0,, and Copyrigt Hougton Mifflin Company. All rigts reserved.
Capter : Functions 9. 8 0 8 0 0 ( ) 0 ( )( + ) 0 Equals 0 at 0 at at 0, and Only 0 and are solutions. 60. + 8 + 8 0 ( + 6) 0 ( + )( ) 0 Equals 0 at 0 at at 0, and Only 0 and are solutions. 6. 6. 6. 6. 6. a. f ( g ) g g( f ) + + + f + 66. a. ( ) ( ) f g g g f f 67. a. f ( g ) g + + g + ( ) g f f 68. a. f ( g ) g + g( f ) f + + Copyrigt Hougton Mifflin Company. All rigts reserved.
Review Eercises 69. f + f( + ) f ( + ) ( + ) + ( + ) ( + + ) ( + ) + ( + ) + + + + + (+ ) + 70. f f( + ) f + ( + ) + ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) 7. Te advertising budget A as a function of t is te composition of A(p) and p(t). ( ()) () ( 8+ ) A( ) 8+ ( ) ( 6) $.6 million 0. 0. A p t p t t 0. 0. 7. a. 0 ( ) 0 ( )( + ) 0 Equals 0 at 0 at at 0, and on [, ] by [, ] 7. a. + 0 ( + ) 0 ( + )( ) 0 Equals 0 at 0 at at 0,, and on [, ] by [, ] 7. a. Te points suggest a parabolic (quadratic) curve. on [0.,.] by [, ] c. For year 6, y 0.7( 6) 0.6( 6) +. $.6 million For year 7, y 0.7( 7) 0.6( 7)+. $.8 million Copyrigt Hougton Mifflin Company. All rigts reserved.