1.1cSOLUTIONS 1
CHAPTER ONE c
c
1. (a)c c Thecstorycinc(a)cmatchescGraphc(IV),cincwhichcthecpersoncforgotchercbookscandchadctocreturnchome.
(b) Thecstorycinc(b)cmatchescGraphc(II),cthecflatctirecstory.cNotectheclongcperiodcofctimecduringcwhichcthecdistancecfrom
chomecdidcnotcchangec(thechorizontalcpart).
(c) Thecstorycinc(c)cmatchescGraphc(III),cincwhichcthecpersoncstartedccalmlycbutcspedcupclater.
Thecfirstcgraphc(I)cdoescnotcmatchcanycofcthecgivencstories.cIncthiscpicture,cthecpersonckeepscgoingcawaycfromchome,
cbutchiscspeedcdecreasescasctimecpasses.cSocacstorycforcthiscmightcbe:cIcstartedcwalkingctocschoolcatcacgoodcpace,cbutcsincec
Icstayedcupcallcnightcstudyingccalculus,cIcgotcmorecandcmorectiredcthecfarthercIcwalked.
2. Thecheightciscgoingcdowncasctimecgoescon.cAcpossiblecgraphciscshowncincFigurec1.1.cThecgraphciscdecreasing.
height
time
Figurec1.1
3. Thecamountcofccarboncdioxideciscgoingcupcasctimecgoescon.cAcpossiblecgraphciscshowncincFigurec1.2.cThecgraphciscincreasing.
CO2
time
Figurec1.2
,2 ChaptercOnec /SOLUTIONS
4. Thecnumbercofcaircconditioningcunitscsoldciscgoingcupcasctemperaturecgoescup.cAcpossiblecgraphciscshowncincFigurec1.3.c
Thecgraphciscincreasing.
ACcunits
temperature
Figurec1.3
5. Thecnoiseclevelciscgoingcdowncascdistancecgoescup.cAcpossiblecgraphciscshowncincFigurec1.4.cThecgraphciscdecreasing.
noiseclevel
distance
Figurec1.4
6. Ifcwecletctcrepresentcthecnumbercofcyearscsincec1900,cthencthecpopulationcincreasedcbetweenctc=c0candctc=c40,cstayedc
approximatelycconstantcbetweenctc=c40candctc=c50,candcdecreasedcforctc≥c50.cFigurec1.5cshowsconecpossiblecgraph.cMan
ycothercanswerscarecalsocpossible.
population
years
20c c 40c c 60c c 80c 100c 120c sincec1900
Figurec1.5
, 1.1cSOLUTIONS 3
7. AmountcofcgrasscGc =c fc(r)cincreasescascthecamountcofcrainfallcrcincreases,csocfc(r)ciscancincreasingcfunction.
8. WecarecgivencinformationcaboutchowcatmosphericcpressurecPc =cfc(ℎ)cbehavescwhencthecaltitudecℎcdecreases:cascaltitudec
ℎcdecreasescthecatmosphericcpressurecPc increases.cThiscmeanscthatcascaltitudecℎcincreasescthecatmosphericcpressurecPc
decreases.cTherefore,cPc =cfc(ℎ)ciscacdecreasingcfunction.
9. WecarecgivencinformationcaboutchowcbatteryccapacitycCc=cfc(Tc)cbehavescwhencairctemperaturecTcdecreases:cascTcdecreases
,cbatteryccapacitycCc alsocdecreases.cThiscmeanscthatcincreasingctemperaturecTc increasescbatteryccapacitycC.cTherefore,
c Cc =cfc(Tc)ciscancincreasingcfunction.
10. TimecTc =c fc(m)cincreasescascmcincreases,csocfc(m)ciscincreasing.
11. ThecattendancecAc=cfc(Pc)cdecreasescascthecpricecPc increases,csocfc(Pc)ciscacdecreasingcfunction.
12. TheccostcofcmanufacturingcCc=cfc(v)cincreasescascthecnumbercofcvehiclescmanufacturedcvcincreases,csocfc(v)ciscancincreasin
gcfunction.
13. Wecarecgivencinformationcaboutchowccommutingctime,cTc =cfc(c)cbehavescascthecnumbercofccarsconcthecroadcccdecreases
:cascccdecreases,ccommutingctimecTc alsocdecreases.cThiscmeanscthatcincreasingcthecnumbercofccarsconcthecroadcccincreas
esccommutingctimecTc.cTherefore,cTc =cfc(c)ciscancincreasingcfunction.
14. Thecstatementcfc(4)c=c20ctellscuscthatcWc =c20cwhenctc=c4.cIncothercwords,cinc2019,cArgentinacproducedc20cmillioncmetri
cctonscofcwheat.
15. (a)c Ifcwecconsidercthecequation
Cc =c 4Tc −c160
simplycascacmathematicalcrelationshipcbetweenctwocvariablescCcandcTc,canycTc valueciscpossible.cHowever,cifcwecthink
cofcitcascacrelationshipcbetweenccricketcchirpscandctemperature,cthencCc cannotcbeclesscthanc0.cSincecCc =c 0cleadscto
c 0c=c4Tc −c160,candcsocTc =c40 F,cwecseecthatcTc cannotcbeclesscthanc40
F.cIncaddition,cwecarectoldcthatcthecfunctionciscnotcdefinedcforctemperaturescabovec134
.cThus,cforcthecfunctioncCc =cfc(Tc)cwechave
Domainc =c AllcTc valuescbetweenc40 Fcandc134 F
=c AllcTc valuescwithc40c ≤cTc ≤c 134
=c[40,c134].
(b) Again,cifcwecconsidercCc=c4Tc−c160csimplycascacmathematicalcrelationship,citscrangeciscallcrealcCcvalues.cHowever
,cwhencthinkingcofcthecmeaningcofcCc =cfc(Tc)cforccrickets,cwecseecthatcthecfunctioncpredictsccricketcchirpscpercminut
ecbetweenc0c(atcTc =c40 F)candc376c(atcTc =c134 F).cHence,
Rangec =c AllcCc valuescfromc0ctoc376
=c AllcCc valuescwithc0c ≤c Cc ≤c376
=c [0,c376].
16. (a)c Thecstatementcfc(19)c=c415cmeanscthatcCc =c415cwhenctc=c19.cIncothercwords,cincthecyearc2019,cthecconcentrationco
fccarboncdioxidecincthecatmospherecwasc415cppm.
(b) Thecexpressioncfc(22)crepresentscthecconcentrationcofccarboncdioxidecincthecyearc2022.
17. (a)c Atcpc =c 0,cwecseecrc =c 8.cAtcpc=c 3,cwecseecrc =c 7.
(b) Whencpc=c2,cwecseecrc=c10.cThus,cfc(2)c=c10.
18. Substitutingc xc =c 5c intocfc(x)c =c 2xc +c 3c gives
fc(5)c=c2(5)c+c3c=c10c+c3c=c13.
19. Substitutingc xc =c 5c intoc fc(x)c =c 10xc −c x2c gives
fc(5)c=c10(5)c−c(5)2c=c50c−c25c=c25.
20. Wecwantcthecy-coordinatecofcthecgraphcatcthecpointcwherecitscx-coordinatecisc5.cLookingcatcthecgraph,cwecseecthatcthe
y-coordinatecofcthiscpointcisc3.cThus
fc(5)c=c3.
, 4 ChaptercOnec /SOLUTIONS
21. Lookingcatcthecgraph,cwecseecthatcthecpointconcthecgraphcwithcancx-coordinatecofc5chascacy-coordinatecofc2.cThus
fc(5)c=c2.
22. Incthectable,cwecmustcfindcthecvaluecofcfc(x)cwhencxc=c5.cLookingcatcthectable,cwecseecthatcwhencxc=c5cwechave
fc(5)c=c4.1
23. (a)c Wecarecaskedcforcthecvaluecofcycwhencxcisczero.cThatcis,cwecarecaskedcforcfc(0).cPluggingcincwecget
fc(0)c=c(0)2c+c2c=c0c+c2c=c2.
(b) Substitutingcwecget
fc(3)c=c(3)2c+c2c =c9c+c2c=c11.
(c) Askingcwhatcvaluescofcxcgivecacy-valuecofc11ciscthecsamecascsolving
yc =c 11c =c x2c +c2
x2c =c 9
,
xc =c ±c c 9c =c ±3.
Weccancalsocsolvecthiscproblemcgraphically.cLookingcatcFigurec1.6,cwecseecthatcthecgraphcofcfc(x)cintersectscthe
c linecyc=c11catcxc=c3candcxc=c−3.cThus,cwhencxcequalsc3corcxcequalsc−3cwechavecfc(x)c=c11.
y
11
x
−6c c −3 3 6
Figurec1.6
(d) No.cNocmattercwhat,cx2ciscgreatercthancorcequalctoc0,csocyc=cx2c+c2ciscgreatercthancorcequalctoc2.
24. Thecyearc2018cwasc2cyearscbeforec2020csoc2018ccorrespondsctoctc=c2.cThus,cancexpressioncthatcrepresentscthecstatementcis:
fc(2)c=c7.088.
25. Thecyearc2020cwasc0cyearscbeforec2020csoc2020ccorrespondsctoctc=c0.cThus,cancexpressioncthatcrepresentscthecstatementcis:
fc(0)cmeters.
26. Thecyearc1949cwasc2020c−c1949c =c 71cyearscbeforec2020csoc1949ccorrespondsctoctc =c 71.cSimilarly,cwecseecthatcthecyear
2000ccorrespondsctoctc=c20.cThus,cancexpressioncthatcrepresentscthecstatementcis:
fc(71)c=cfc(20).
27. Thecyearc2018cwasc2cyearscbeforec2020csoc2018ccorrespondsctoctc=c2.cSimilarly,ctc=c3ccorrespondsctocthecyearc2017.cThus,
cfc(3)candcfc(2)carecthecaveragecannualcseaclevelcvalues,cincmeters,cinc2017candc2018,crespectively.cBecausec11cmillimeter
sciscthecsamecasc0.011cmeters,cthecaveragecseaclevelcinc2018,cfc(2),cisc0.011clesscthancthecseaclevelcinc2017cwhichciscfc(3).c
Ancexpressioncthatcrepresentscthecstatementcis:
fc(2)c=cfc(3)c−c0.011.
Notecthatctherecarecothercpossiblecequivalentcexpressions,csuchcas:cfc(2)c−cfc(3)c =c−0.011.
CHAPTER ONE c
c
1. (a)c c Thecstorycinc(a)cmatchescGraphc(IV),cincwhichcthecpersoncforgotchercbookscandchadctocreturnchome.
(b) Thecstorycinc(b)cmatchescGraphc(II),cthecflatctirecstory.cNotectheclongcperiodcofctimecduringcwhichcthecdistancecfrom
chomecdidcnotcchangec(thechorizontalcpart).
(c) Thecstorycinc(c)cmatchescGraphc(III),cincwhichcthecpersoncstartedccalmlycbutcspedcupclater.
Thecfirstcgraphc(I)cdoescnotcmatchcanycofcthecgivencstories.cIncthiscpicture,cthecpersonckeepscgoingcawaycfromchome,
cbutchiscspeedcdecreasescasctimecpasses.cSocacstorycforcthiscmightcbe:cIcstartedcwalkingctocschoolcatcacgoodcpace,cbutcsincec
Icstayedcupcallcnightcstudyingccalculus,cIcgotcmorecandcmorectiredcthecfarthercIcwalked.
2. Thecheightciscgoingcdowncasctimecgoescon.cAcpossiblecgraphciscshowncincFigurec1.1.cThecgraphciscdecreasing.
height
time
Figurec1.1
3. Thecamountcofccarboncdioxideciscgoingcupcasctimecgoescon.cAcpossiblecgraphciscshowncincFigurec1.2.cThecgraphciscincreasing.
CO2
time
Figurec1.2
,2 ChaptercOnec /SOLUTIONS
4. Thecnumbercofcaircconditioningcunitscsoldciscgoingcupcasctemperaturecgoescup.cAcpossiblecgraphciscshowncincFigurec1.3.c
Thecgraphciscincreasing.
ACcunits
temperature
Figurec1.3
5. Thecnoiseclevelciscgoingcdowncascdistancecgoescup.cAcpossiblecgraphciscshowncincFigurec1.4.cThecgraphciscdecreasing.
noiseclevel
distance
Figurec1.4
6. Ifcwecletctcrepresentcthecnumbercofcyearscsincec1900,cthencthecpopulationcincreasedcbetweenctc=c0candctc=c40,cstayedc
approximatelycconstantcbetweenctc=c40candctc=c50,candcdecreasedcforctc≥c50.cFigurec1.5cshowsconecpossiblecgraph.cMan
ycothercanswerscarecalsocpossible.
population
years
20c c 40c c 60c c 80c 100c 120c sincec1900
Figurec1.5
, 1.1cSOLUTIONS 3
7. AmountcofcgrasscGc =c fc(r)cincreasescascthecamountcofcrainfallcrcincreases,csocfc(r)ciscancincreasingcfunction.
8. WecarecgivencinformationcaboutchowcatmosphericcpressurecPc =cfc(ℎ)cbehavescwhencthecaltitudecℎcdecreases:cascaltitudec
ℎcdecreasescthecatmosphericcpressurecPc increases.cThiscmeanscthatcascaltitudecℎcincreasescthecatmosphericcpressurecPc
decreases.cTherefore,cPc =cfc(ℎ)ciscacdecreasingcfunction.
9. WecarecgivencinformationcaboutchowcbatteryccapacitycCc=cfc(Tc)cbehavescwhencairctemperaturecTcdecreases:cascTcdecreases
,cbatteryccapacitycCc alsocdecreases.cThiscmeanscthatcincreasingctemperaturecTc increasescbatteryccapacitycC.cTherefore,
c Cc =cfc(Tc)ciscancincreasingcfunction.
10. TimecTc =c fc(m)cincreasescascmcincreases,csocfc(m)ciscincreasing.
11. ThecattendancecAc=cfc(Pc)cdecreasescascthecpricecPc increases,csocfc(Pc)ciscacdecreasingcfunction.
12. TheccostcofcmanufacturingcCc=cfc(v)cincreasescascthecnumbercofcvehiclescmanufacturedcvcincreases,csocfc(v)ciscancincreasin
gcfunction.
13. Wecarecgivencinformationcaboutchowccommutingctime,cTc =cfc(c)cbehavescascthecnumbercofccarsconcthecroadcccdecreases
:cascccdecreases,ccommutingctimecTc alsocdecreases.cThiscmeanscthatcincreasingcthecnumbercofccarsconcthecroadcccincreas
esccommutingctimecTc.cTherefore,cTc =cfc(c)ciscancincreasingcfunction.
14. Thecstatementcfc(4)c=c20ctellscuscthatcWc =c20cwhenctc=c4.cIncothercwords,cinc2019,cArgentinacproducedc20cmillioncmetri
cctonscofcwheat.
15. (a)c Ifcwecconsidercthecequation
Cc =c 4Tc −c160
simplycascacmathematicalcrelationshipcbetweenctwocvariablescCcandcTc,canycTc valueciscpossible.cHowever,cifcwecthink
cofcitcascacrelationshipcbetweenccricketcchirpscandctemperature,cthencCc cannotcbeclesscthanc0.cSincecCc =c 0cleadscto
c 0c=c4Tc −c160,candcsocTc =c40 F,cwecseecthatcTc cannotcbeclesscthanc40
F.cIncaddition,cwecarectoldcthatcthecfunctionciscnotcdefinedcforctemperaturescabovec134
.cThus,cforcthecfunctioncCc =cfc(Tc)cwechave
Domainc =c AllcTc valuescbetweenc40 Fcandc134 F
=c AllcTc valuescwithc40c ≤cTc ≤c 134
=c[40,c134].
(b) Again,cifcwecconsidercCc=c4Tc−c160csimplycascacmathematicalcrelationship,citscrangeciscallcrealcCcvalues.cHowever
,cwhencthinkingcofcthecmeaningcofcCc =cfc(Tc)cforccrickets,cwecseecthatcthecfunctioncpredictsccricketcchirpscpercminut
ecbetweenc0c(atcTc =c40 F)candc376c(atcTc =c134 F).cHence,
Rangec =c AllcCc valuescfromc0ctoc376
=c AllcCc valuescwithc0c ≤c Cc ≤c376
=c [0,c376].
16. (a)c Thecstatementcfc(19)c=c415cmeanscthatcCc =c415cwhenctc=c19.cIncothercwords,cincthecyearc2019,cthecconcentrationco
fccarboncdioxidecincthecatmospherecwasc415cppm.
(b) Thecexpressioncfc(22)crepresentscthecconcentrationcofccarboncdioxidecincthecyearc2022.
17. (a)c Atcpc =c 0,cwecseecrc =c 8.cAtcpc=c 3,cwecseecrc =c 7.
(b) Whencpc=c2,cwecseecrc=c10.cThus,cfc(2)c=c10.
18. Substitutingc xc =c 5c intocfc(x)c =c 2xc +c 3c gives
fc(5)c=c2(5)c+c3c=c10c+c3c=c13.
19. Substitutingc xc =c 5c intoc fc(x)c =c 10xc −c x2c gives
fc(5)c=c10(5)c−c(5)2c=c50c−c25c=c25.
20. Wecwantcthecy-coordinatecofcthecgraphcatcthecpointcwherecitscx-coordinatecisc5.cLookingcatcthecgraph,cwecseecthatcthe
y-coordinatecofcthiscpointcisc3.cThus
fc(5)c=c3.
, 4 ChaptercOnec /SOLUTIONS
21. Lookingcatcthecgraph,cwecseecthatcthecpointconcthecgraphcwithcancx-coordinatecofc5chascacy-coordinatecofc2.cThus
fc(5)c=c2.
22. Incthectable,cwecmustcfindcthecvaluecofcfc(x)cwhencxc=c5.cLookingcatcthectable,cwecseecthatcwhencxc=c5cwechave
fc(5)c=c4.1
23. (a)c Wecarecaskedcforcthecvaluecofcycwhencxcisczero.cThatcis,cwecarecaskedcforcfc(0).cPluggingcincwecget
fc(0)c=c(0)2c+c2c=c0c+c2c=c2.
(b) Substitutingcwecget
fc(3)c=c(3)2c+c2c =c9c+c2c=c11.
(c) Askingcwhatcvaluescofcxcgivecacy-valuecofc11ciscthecsamecascsolving
yc =c 11c =c x2c +c2
x2c =c 9
,
xc =c ±c c 9c =c ±3.
Weccancalsocsolvecthiscproblemcgraphically.cLookingcatcFigurec1.6,cwecseecthatcthecgraphcofcfc(x)cintersectscthe
c linecyc=c11catcxc=c3candcxc=c−3.cThus,cwhencxcequalsc3corcxcequalsc−3cwechavecfc(x)c=c11.
y
11
x
−6c c −3 3 6
Figurec1.6
(d) No.cNocmattercwhat,cx2ciscgreatercthancorcequalctoc0,csocyc=cx2c+c2ciscgreatercthancorcequalctoc2.
24. Thecyearc2018cwasc2cyearscbeforec2020csoc2018ccorrespondsctoctc=c2.cThus,cancexpressioncthatcrepresentscthecstatementcis:
fc(2)c=c7.088.
25. Thecyearc2020cwasc0cyearscbeforec2020csoc2020ccorrespondsctoctc=c0.cThus,cancexpressioncthatcrepresentscthecstatementcis:
fc(0)cmeters.
26. Thecyearc1949cwasc2020c−c1949c =c 71cyearscbeforec2020csoc1949ccorrespondsctoctc =c 71.cSimilarly,cwecseecthatcthecyear
2000ccorrespondsctoctc=c20.cThus,cancexpressioncthatcrepresentscthecstatementcis:
fc(71)c=cfc(20).
27. Thecyearc2018cwasc2cyearscbeforec2020csoc2018ccorrespondsctoctc=c2.cSimilarly,ctc=c3ccorrespondsctocthecyearc2017.cThus,
cfc(3)candcfc(2)carecthecaveragecannualcseaclevelcvalues,cincmeters,cinc2017candc2018,crespectively.cBecausec11cmillimeter
sciscthecsamecasc0.011cmeters,cthecaveragecseaclevelcinc2018,cfc(2),cisc0.011clesscthancthecseaclevelcinc2017cwhichciscfc(3).c
Ancexpressioncthatcrepresentscthecstatementcis:
fc(2)c=cfc(3)c−c0.011.
Notecthatctherecarecothercpossiblecequivalentcexpressions,csuchcas:cfc(2)c−cfc(3)c =c−0.011.
