Chapter 1 Functions and Graphs
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- Alannah Linette Price
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1 Capte Functions and Gaps Section n n n n n n n 6 n 6 n n a b ± ± 6 c ± 6 ± 8 8 o y 8y 7 8y y 8y y 8 o y y. 7 7 o 7 7
2 Capte : Functions and Gaps. [ ][ ] 8 o ± 6 ± o. ± ± ± o. ± ± 6 8 ± 8 ± ±, o. ± ± 6 ± ± 6 6 6, o ± ± ±, o 7. a b c ± ± 8 o. a b ± c. < < 8 <,. > 6 > < 6, 6 ± ±
3 Section , ] , > > 8, > 8 <. > > 8, > 8 <. 7 > 7 > Te poduct is positive. Te citical values ae and < < Te poduct is negative. Te citical values ae and.., 7, Te poduct is positive o zeo. Te citical values ae and 7. 7,, ] [7, Te poduct is negative o zeo. Te citical values ae and.. < < <,,
4 Capte : Functions and Gaps 6. < < < 8 < < 8, > < < o >, 7, 6. > 7 6. > < o > < > < >,, 7. o 8, 8] [, o, 8, 8 ], 7. Because an absolute value is al-ways nonnegative, te inequality is always tue. Te solution set consists of all eal numbes., 77. A A LW LW L W P 7 W 7 W 7 W 7 W W 7W 7 P L W L W 7 W 7 W W 7 7 L 7 7L L o W LW L. L Te ectangle measues. cm by cm. 7. Because an absolute value is always nonnegative, te inequality < as no solution. Tus te only solution of te inequality is te solution of te equation. 7. A lw P 6 l w l w l w 6 w 6 w, w 6w w 6w, w w, w w w ft l ft Te dimensions ae feet by feet.
5 Section. 8. Plan A:. Plan B:.8. <.8 <.7 7. < Plan A is less epensive if you use at least 8 cecks. 8. Plan A: 8 Plan B:. 8 >.. > >. Plan A pays bette if at least sales ae made F 68 C 6 C C 7... Connecting Concepts 87. n n n n 6 n n n So L. 8. R > > Te poduct is positive. Te citical values ae and.. s 6t vt s s > 8, v 6, s 6t 6t > 8 6t 6t 8 > 6 t t > 6 t t > Te poduct is positive. Te citical values ae t and t. 6 t t,. a. s.. b. s.., o s.. s.6 s. citical values. s.6 second < t < seconds Te ball is ige tan 8 ft between and seconds.... Pepae fo Section y? No, te equation is not tue. 7. o
6 6 Capte : Functions and Gaps Section.. a. b aveage Te aveage incease in eat ate is. beats pe minute.. d d d d d a a bb a b a b a b a b. d wit < Note: since <,
7 Section o Te points ae,,,.. y, y. M M 6 6,, 8, 6, 6,, M,.7 7.8,.87, ,,,,, Intecepts:,, 6,.,,,,,., ±, ±, y 7., ±, ±, y y y. cente,, adius 6. cente,, adius 7. cente,, adius. cente 8,, adius
8 8 Capte : Functions and Gaps 7. y. y 6. y 6 y 6. y y 6 y 67. y 8 y 6 y 8y y cente 7,, adius 7. y y y y y cente,, adius 6. 6 y 6 y y cente,, adius 6. y 6 6 y 6 y y 6 y cente,, adius 7. d 6 6 Since te diamete is, te adius is. Te cente is te midpoint of te line segment fom, to -,. -,,7 cente y 7 7. Since it is tangent to te -ais, its adius is. 7 y
9 Section.... Connecting Concepts y,, teefoe 8 and y y 6 y 8. y 8,, 7 teefoe and 7 y 8 7 y 8 y 6 Tus 7, 6 is te ote endpoint. Tus, is te ote endpoint.. y y 6 6 8y y 6 y 8y. y y y y y 6 8 y y 6 8 y 6 y y 6 6 y y 6 6 y Simplifying yields y.. Te cente is -,. Te adius is. y... Pepae fo Section D {,,,,} R {,,,}
10 Capte : Functions and Gaps. d 8. 6, a, a 6 6 a Section.. Given f, a. f 6 b. f c. f d. f e. f k k k f. f k k k 6 k. Given A w w, a. b. c. d. e. f. A A A A A 6 A c c c. Given f, a. b. f f c. f d. f f e. f c c c f. f 7. Given s, a. s b. s c. s d. s e. Since t >, t t. s t t t t t f. Since t <, t t. t t s t t t. a. Since <, use P. P b. Since, use P. 6 P c. Since c <, use P. P c c d. Since k, ten k, so use P. P k k k k k k k k
11 Section.. y 7 y 7 y 7, y is a function of.. y ±, y is not a function of since fo eac > tee ae two values of.. y. y y y ±, y is a not function of. 7. y, y is a function of.. Function; eac is paied wit eactly one y. y±, y is a not function of.. Function; eac is paied wit eactly one y.. Function; eac is paied wit eactly one y. 7. f Domain is te set of all eal numbes.. f Domain is te set of all eal numbes.. f Domain is { }.. f 7 Domain is { 7}.. f Domain is { }. 7. f Domain is { > }... Domain: te set of all eal numbes Domain: te set of all eal numbes.. Domain: { 6 6} Domain: { } 7. a. C.8.7.int.8.7.int $. b. cw. a. Yes; evey vetical line intesects te gap in one point. b. Yes; evey vetical line intesects te gap in one point. c. No; some vetical lines intesect te gap at moe tan one point. d. Yes; evey vetical line intesects te gap in at most one point.. Deceasing on, ] ; inceasing on [,. Inceasing on,
12 Capte : Functions and Gaps. Deceasing on, ] ; inceasing on [, ] ; deceasing on [, ] ; inceasing on [, 7. Constant on, ] ; inceasing on [,. Deceasing on, ] ; constant on [, ]; inceasing on [, 6. g and F ae one-to-one since evey oizontal line intesects te gap at one point. f, V, and p ae not one-to-one since some oizontal lines intesect te gap at moe tan one point. 6. a. l w w l w l 6. v t 8, 6t, t b. A lw A l l A ll 67. a. C.8.8 b. R 7. c. P 7. C 7. [.8 ] Note is a natual numbe. 7. d t d t metes, t 6 7. d 8t 6t miles wee t is te numbe of ous afte : noon 7. a. b. Cicle Squae C π C s π s s π Aea π π Aea s π π 6 Total Aea π 6 π Total Aea a. b. Left side tiangle c c Rigt side tiangle c c Total lengt Total Lengt c. Domain:, c. Domain: [, ].
13 Section. 7.. Y answes accuate to neaest apple 8. f c c c c c 6 c c c o c c c 8. is not in te ange of f, since only if o. 8. Set te gaping utility to dot mode. WINDOW FORMAT Xmin-.7 Xma.7 Xscl Ymin- Yma Yscl 87. WINDOW FORMAT Xmin-.7 Xma.7 Xscl Ymin- Yma Yscl Connecting Concepts. f 6. f 6. a. f,7 7 6 b. f, c. f, d. f, e. f k,k k k k f. f k, k k k k 6 k 8k 7. s 8 A,8, a a a a a a a a o a
14 Capte : Functions and Gaps.... Pepae fo Section.. d 7. Te poduct of any numbe and its negative ecipocal is y y y 6. y y 6 y 8. y Section.. y y 7 m. m. Te line does not ave a slope since. 7. m 6 6. m m f f f f. m f f f f. m y-intecept, 7. m y-intecept,. m y-intecept,. m y-intecept,
15 Section.. m y-intecept,. m y-intecept, 7. Use y m b wit m, b. y. Use y m b wit m, b. y. Use y m b wit m, b. y. y y y. m y y y 7. m 7 y 7 8 y 8 y. f. f. f. f Te -intecept of te gap of f is,. Xmin, Xma 6, Xscl, Ymin., Yma, Yscl
16 6 Capte : Functions and Gaps 7. f Te -intecept of te gap of f is,.. Algebaic metod: f f 6 Gapical metod: Gap y and y 6 Tey intesect at, y 6. Xmin, Xma, Xscl, Ymin, Yma, Yscl. Algebaic metod: f f 6 6 Gapical metod: Gap y y and Tey intesect at, y 6. Xmin 7.8, Xma 7.8, Xscl, Ymin,Yma, Yscl. m Te value of te slope indicates tat te speed of sound in wate inceases.87 ft pe s fo a one-degee incease in tempeatue. Xmin, Xma, Xscl, Ymin, Yma, Yscl. a. m. Hc. c Hc.c 7. a. 6,8, m Nt 6, t Nt t,6, b. H mpg b. 6, t,6,,, t 8.8 t Te numbe of jobs will eceed 6, in 8.
17 Section. 7. a. m 8 86 Bd 8 d6 Bd d b. Te value of te slope means tat a -inc incease in te diamete of a log ft long esults in an incease of boad-feet of lumbe tat can be obtained fom te log. c. B 7 boad feet 6. Line A epesents Micelle Line B epesents Amanda Line C epesents te distance between Micelle and Amanda. 6. a. Find te slope of te line. 8 7 m Use te point-slope fomula to find te equation. y y m y.8 7 y.8 8. y.8 8. b. y.8 8. y y P 8,7 P 8,7 P 7,7 7, , te beak-even point 6. P. 78 P.78 P , te beak-even point 6. a. C $ 7 b. C $ 8 c. C $ d. Te maginal cost is te slope of C 8 7, wic is 8 dollas. 7. a. C t,. 6. 7t b. R t. t c. Pt Rt Ct Pt. t,. 6.7 t Pt.t,.6.7t Pt 8.t,. d. 8.t,.,. t 8. t. days days 7. Te gap of y as m. y y y
18 8 Capte : Functions and Gaps 7. Te gap of y as m. Tus we use a slope of. y y y 77. Te equation of te line toug, and P, as slope. Te pat of te ock is on te line toug P, wit slope, so y. y y y Te point wee te ock its te wall at y is te point of intesection of y and y. feet Teefoe te ock its te wall at,. Te -coodinate is. 7. a. [ ] so Q, Q, Q, m b. [ ].so Q, Q.,. Q.,. m..... c. [ ].so Q, Q.,. Q.,... m... d. As appoaces, te slope of PQ seems to be appoacing. e. [ ], y,, y [ ] y y m 8. m
19 Section.... Connecting Concepts 8. y Substitute y y fo m in te point-slope fom y y m to yield y y y, te two-point fom. 8. y y y y y y 87. y Use wit a and b. a b y y y 8. y Use wit b a. a b y Since, is on te line, a a a a a a a a a 7 a 7 a y Tus 7 7 y 7 7 y 7.. y Te slope of te line toug, and, y is, so. Teefoe y y 8 y 7 Substitute y into tis equation. 7 o 8 8 If, y,. If, y,, but tis is te point itself. 8 Te point, is on te gap of y, and te slope of te line containing, and 8, is.
20 Capte : Functions and Gaps... Pepae fo Section f ± ± t 6t 6t 6t 8 t t t t t, Section.. d. b. g 7. c. f standad fom, vete,, ais of symmety. f standad fom, vete,, ais of symmety. f vete standad fom,,, ais of symmety. f 6 standad fom, vete, 6, ais of symmety
21 Section. 7. f standad fom, vete,, ais of symmety. b a y f vete, f. b a y f vete, f. b 6 6 a y f 6 8 vete, f. b a 7. b a 8 y f vete, 8 7 f 8 y f vete, f 8 6
22 Capte : Functions and Gaps. f vete, Te y-value of te vete is. Te paabola opens up since a >. Tus te ange is { y y }. f o. f vete,. f vete, 8 7 Te y-value of te vete is. 8 Te paabola opens down since a <. 7 Tus te ange is y y. 8 f o. f minimum value of 6 wen Te y-value of te vete is. Te paabola opens up since a >. Tus te ange is y y. No, y y.
23 Section. 7. f maimum value of wen. f minimum value of wen. f minimum value of wen 8. f maimum value of wen a. Te maimum eigt of te ac is 7 feet. b feet 6 6 c wen.feet 7. a. w l 6 w 6l w 6l b. A w l c. 6 l A l l l A l l A l l, In standad fom, A l, Te maimum aea of, ft is poduced wen 6 l ft and te widt w ft.
24 Capte : Functions and Gaps. a. T t.7t.t...7 t t..7.7 t t t t t t 6 7 Te tempeatue is a maimum wen 7 t 6 ous afte 6: A.M. 7 7 Note 7 6 minutes minutes. Tus te tempeatue is a maimum at : P.M. b. Te maimum tempeatue is appoimately F.. Nt.t.t 7.68 Nt. t 8 t 7.68 Nt. t.8 minimum at t, o fo omes > Solve fo using quadatic fomula... 8 ± ± 6,, use positive value of 6. Yes, te conditions ae satisfied.. a. Ev.8v.76v v v..8.8 v 8 v. v v v.68 Te maimum fuel efficiency is obtained at a speed of mp. b. Te maimum fuel efficiency fo tis ca, to te neaest mile pe gallon, is mpg. 7. Let y, ten 6 6 o 6 6 Te -intecepts ae, and 6,. Let, ten f 6 Te y-intecept is,.. Let y, ten 6 ± 6 Since te disciminant 6 7 is negative, tee ae no -intecepts. Let, ten f 6 6 Te y-intecept is, b a b a.7 8. R , P Tus, 7 units yield a maimum evenue of $,. Tus, 8 units yield a maimum pofit of $..
25 Section. 6. P R C Te beak-even points occu wen R C o P. Tus,. 8 ±. 8. ± 76. ±. o 6 Te beak-even points occu wen o Let te numbe of people tat take te tou. a. R b. P R C b 7. c. a. P $76. d. Te maimum pofit occus wen. 6. t 6t 8t b 8 a. seconds a 6 b feet c. 6t 8t 6 tt 8 6t o t 8 t t 8 Te pojectile its te gound at t 8 seconds. 7. y.. b a... y feet
26 6 Capte : Functions and Gaps 7. Te peimete is 8. π Solve fo. π 8 π 8 Aea semicicle ectangle A 8 π 8 π π π 8 π π 8 π π 8 π π Gap te function A to find tat its maimum occus wen 6.7 feet. Xmin, Xma, Xscl Ymin, Yma, Yscl 8 π 8 π feet Hence te optimal window as its semicicula adius equal to its eigt. Note: Using calculus it can be sown tat te eact value of. π 8
27 Section Connecting Concepts 7. f a b ab a. -intecepts occu wen y. a b ab a b a o b a b Tus te -intecepts ae a, and b,. b a b a b b. wic is te -coodinate of a te midpoint of te segment joining a, and b,. 77. Let f a b c. We know f a b c f a b c Tis implies c and fom Equation we ave a b o a b Te -value of te vete is, and by te vete fomula we ave b a, wic implies b a. Substituting a fo b in Equation gives us a a a 8a a a Substituting fo a in Equation gives us b b b 6 b Tus te desied quadatic function is f. 7. P w 6 w a. w 6 b. Aea A w A 6 A 6 8. Te disciminant is b b, wic is always positive. Tus te equation as two eal zeos fo all values of b. 8. Inceasing te constant c inceases te eigt of eac point on te gap by c units. 8. Let one numbe. Ten 8 te ote numbe. P 8 8, vete at. Tus, and 8.Te numbes ae and. b a 8 87., y,, y y y m
28 8 Capte : Functions and Gaps... Pepae fo Section.6 8. f 6 b a. f 6 6 f [ ] [6 ] 6 f f. f. f. f f. f g [ ] f g [ ] f g [ ] f g [ ] f g [ ]. a a, b b b midpoint is, b. a a, b b midpoint is, Section Replacing by leaves te equation unalteed. Tus te gap is symmetic wit espect to te y-ais.. Not symmetic wit espect to eite ais. neite 7. Symmetic wit espect to bot te - and te y-aes.. Symmetic wit espect to bot te - and te y-aes.. Symmetic wit espect to bot te - and te y-aes.. No, since y simplifies to y, wic is not equivalent to te oiginal equation y.. Yes, since y implies y o y, wic is te oiginal equation. 7. Yes, since y simplifies to te oiginal equation.. Yes, since y simplifies to te oiginal equation.
29 Section.6... symmetic wit espect to te y-ais symmetic wit espect to te oigin symmetic wit espect to te oigin 7... symmetic wit espect to te line symmetic wit espect to te line no symmety. Even since g 7 7 g.. Odd, since F F. 7. Even. Even. Even. Even. Neite 7.. a. f b. f
30 Capte : Functions and Gaps 6. a. f,,,,,, 6. a. f b. f,,6,,,, b. f 6. a. f,,, 67. b. f,,, a. b.
31 Section.6 7. a. 7. b a. b. c.... Connecting Concepts 8. a. b. f f
32 Capte : Functions and Gaps... Pepae fo Section f a a a 8a a. f 8 8. Domain: all eal numbes ecept. 8 Domain: > Section.7. f g Domain all eal numbes f g 8 Domain all eal numbes f g Domain all eal numbes f / g / { } Domain. f g 8 Domain all eal numbes f g 8 Domain all eal numbes f g 8 6 Domain all eal numbes f / g 8/ [ ]/ Domain { }. f g 7 8 Domain all eal numbes f g 7 6 Domain all eal numbes f g 7 7 Domain all eal numbes f / g 7 / 7 Domain { } 7. f g 7 7 Domain all eal numbes f g 7 Domain all eal numbes f g Domain all eal numbes f / g 7/ Domain,
33 Section.7. g f Domain { } f g Domain { } f g Domain { } f / g Domain { }. f g Domain { } f g Domain { } f g Domain { } f / g Domain { }. f g f g 8. f g f g 7. f g 6 f g 6 6. f g 6 f g 6 6. fg fg fg 6 8 fg f g f g f g o 7. f g f g. f f [ ]
34 Capte : Functions and Gaps. [ ] f f f f. f f go f g[ f ] g[ ] f o g f [ g ] f [ 7] [ ] [ ] g f g o f o g f [ ] [ ] [ ] 8 8. go f g[ f ] f o g f [ g ] g f [ ] [ ] [ ]. go f g[ f ] f o g f [ g ] f [ ] g 6 6 [ ]
35 Section.7. go f g[ f ] f o g f [ g ] f g 7. go f g f og f [ ] Use te esults to wok Eecises to 6.. go f 6 go f go go 7. og 7 7 og go f 6 go f c c c 6 6c c6. f og f o g 8. f o f f o f8 8. go f 6 go f go go k k k k k 6k k k 8k k 6k k 6k k 8k k 6k k 8k
36 6 Capte : Functions and Gaps 6. a..t and A π [ t] so At π 67. a. Since d s, d s 6 π. t. π π squae feet 8.7 squae feet b.. t. t t and V π so Vt π. t [ t ].π t Note: V π π A t. πt. πt V. π 6.7 π cubic feet.8 cubic feet b. d s 6 d 8 t 6 s 8t 6t t 6 t 6t 88 s 8 d ft 6. Y o F Y F convets inces to yads. F takes inces to feet, and ten Y takes feet to yads. 7. a. On [, ], a t C a t C.8 C a C C C Aveage ate of cange.8. 8 Tis is identical to te slope of te line toug C C, C and, C since m C C b. On [,. ], a t. C. C 78. Aveage ate of cange 6... c. On [, ], a t C C..8 Aveage ate of cange.7 d. On [,. ], a t.. C. C Aveage ate of cange.8...
37 Section e. On [,. ], a t.. C. C.7.8. Aveage ate of cange 6...., t, t t t t f. On [ ] Con Con t t t t t 7 t 7 t t t t 7 t t t Con t Con 7 t t Aveage ate of cange t t 7 t t t 7 t t As t appoaces, te aveage ate of cange ove [, t] seems to appoac.... Connecting Concepts 7. go f g[ f ] f o g f [ g ] g[ ] f [ ] 7 7 go f f o g 7. go f g[ f ] f og f [ g ] 6 g f go f f o g 77. go f g[ f ] f o g f [ g ] g[ ] f [ ]
38 8 Capte : Functions and Gaps 7. go f g[ f ] g [ ] f og f g f 8. go f g[ f ] f o g f [ g ] g f... Capte Tue/False Eecises. False. Let f.ten f f, but.. False. Let f and g. Ten f o g f, but g o f g.. Tue. Tue. 6. False. Let f. [ f ] [], weeas f [ f ] f. False. Let f f. Ten. f 7. Tue 8. False. Let f. Ten f f, weeas f f.. Tue. Tue. Tue. Tue. Tue... Capte Review. z [.] z z. y 6y [.] y 8 y. [.] 6 6
39 Capte Review. m m [.] 6m m 6m m m m. y y 8 [.] y 6 y y 6 o y y 6 o y 6. z z z z z z z o [.] z z 7. v v v v ± v ± v 6 [.] 8. s s [.] s s ± s ± s ± s. c c 7 [.] c c 6. 7a > a 7a > 6a 8 a > [.] a >., ] [, [.] Citical values ae and.. < < < Citical values ae < < [.] and.. > [.] > o < > 8 < > <,,. [.]. d 7 [.] 6. d 8 [.] ,,, [.] ,,, [.]. cente,, adius [.]. y y y y y cente,, adius [.]
40 Capte : Functions and Gaps. y [.]. y 8 8 y 8 [.]. a. f [.] b. f 7 c. f t t t d. f 6 e. f t t t t t f. f t t t t t 7t t. a. b. c. d. e. f. g 6 [.] 6 g 6 6 g g 6 6 g t 6 t g t 6 t 6t 6 t 6t
41 Capte Review. a. f o g f[ g] [.7] f [ 8] f [ ] b. g o f g[ f ] g[ ] g[ ] g[ ] [ ] 8 c. f o g f[ g ] f[ 8] d. g o f g[ f ] g[ ] [ ] f f [.7] a. b. c. d. f o g f[ g ] [.7] f [ ] f [6] g o f g[ f ] g[ 7] g[7] 7 6 f o g f[ g ] [ ] f g o f g[ f ] g[ 7] g g [.7]
42 Capte : Functions and Gaps.. f is inceasing on [, f is deceasing on, ] [.] f is inceasing on [, f is deceasing on, ] [.].. f is inceasing on [, ] f is constant on, ] [, [.] f is constant on..., [6,, [,, [,, [,, [,, [,, [,,... [.].. f is inceasing on, [.] f is inceasing on, [.]. Domain { is a eal numbe} [.] 6. Domain { 6} [.] 7. Domain { } [.] 8. Domain {, } [.]. 7 m [.] y y y point - slope fom. m [.] 7 7 y 7 y 7. y 8 y 8 y [.] Slope of paallel line is. y y y y y. y [.] y y Slope of pependicula line is. y 7 [ ] y 7 y 7 y 7 y y
43 Capte Review. 6 6 f f f [.]. f [.] f f f. 8 f [.] f f f 6. 6 f [.] 6 f f f f 7. f [.] f f f f 8. 6 f [.] 6 6 f f f a b [.] f Tus te vete is, 8.. a b [.] f Tus te vete is, a b [.] f Tus te vete is, a b [.] 6 8 f Tus te vete is,.
44 Capte : Functions and Gaps.. m b y d, y,, y, [.] m d d d d. a. Revenue b. Pofit Revenue Cost P. P. P. c. Beak even Revenue Cost.. 8 Te company must sip 8 pacels. [.] 6. [.6] [.6] Te gap of y 7 is symmetic wit espect to te y-ais. [.6] Te gap of y is symmetic wit espect to te -ais. [.6] Te gap of y is symmetic wit espect to te oigin. [.6] Te gap of y is symmetic wit espect to te -ais, y-ais, and te oigin. [.6] Te gap of y is symmetic wit espect to te -ais, y-ais, and te oigin. [.6] 6. Te gap of y 8 is symmetic wit espect to te oigin. [.6] 6. Te gap of y is symmetic wit espect to te -ais, y-ais, and te oigin. [.6] 6. Te gap of y is symmetic wit espect to te oigin. [.6] a. Domain all eal numbes y y Range { } b. g is an even function [.6] a. Domain all eal numbes Range all eal numbes b. g is neite even no odd [.6]
45 Capte Review a. Domain all eal numbes y y Range { } b. g is an even function [.6] a. Domain { } Range { y y } b. g is an even function [.6] a. Domain all eal numbes Range all eal numbes b. g is an odd function [.6] a. Domain all eal numbes y y is an even intege Range { } b. g is neite even no odd [.6] 7. F 7 [.6] F 7 F 7 F 7. A 6 [.6] A 6 A 6 A 7. P [.6] P 7. G 8 [.6] G G 8 G
46 6 Capte : Functions and Gaps W [.6] W W W W 76. T [.6] T T T 77. [.6] 78. [.6] 7. [.6] 8. [.] 8. [.] 8. [.]
47 Capte Review 7 8. f g [.7] 6 Te domain is all eal numbes. f g Te domain is all eal numbes. fg 7 Te domain is all eal numbes. f g Te domain is{. } 8. f g 8 [.7] Te domain is all eal numbes. f g 8 8 Te domain is all eal numbes. fg Te domain is all eal numbes. f 8 g Te domain is esticted wen. ± ± 6 ± wic is not a eal numbe Teefoe te domain is all eal numbes. 8. Let one of te numbes and te ote numbe. Tei poduct is given by y. Now y takes on its maimum value wen b. a Tus te two numbes ae and. Tat is, bot numbes ae. [.] 86. Let te smalle numbe. Let equal te lage numbe. Te sum of tei squaes y is given by y Now y takes on its minimum value wen b a Tus te numbes ae and. [.]
48 8 Capte : Functions and Gaps 87. s t t [.] a. Aveage velocity ft/sec b. Aveage velocity 7 ft/sec. c. Aveage velocity ft/sec.. d. Aveage velocity ft/sec. e. It appeas tat te aveage velocity of te ball appoaces ft/sec. 88. s t t t [.] [ ] a. Aveage velocity [ ] [8 ] 8 7 ft/sec [ ] b. Aveage velocity 6 [ ] [8 ] 8 ft/sec.. [ ] c. Aveage velocity... [ ]... [8 ] ft/sec... [ ] d. Aveage velocity..6. [ ]. 8.. [8 ] ft/sec e. It appeas tat te aveage velocity of te ball appoaces ft/sec.
49 Capte Test... Capte Test. 8 [.]. [.] ± ± ± [.]. [.] 6 Citical values ae 6 and. 6. midpoint y, y,,, [.] lengt d y y y [.] y Tus te -intecept is,. y y y ± y Tus te y-intecepts ae, and,. 7. y [.] 8. y y y y y y y cente,, adius [.]. 6 Te poduct is positive o zeo. Te citical values ae and.. Te domain is { o }. [.] a. inceasing on, ] b. neve constant c. deceasing on [, [.]
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