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The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education

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ALGEBRA II

An Integrated Approach

ALGEBRA II // MODULE 1

FUNCTIONS AND THEIR INVERSES

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 1 - TABLE OF CONTENTS

FUNCTIONS AND THEIR INVERSES

1.1Brutus Bites Back – A Develop Understanding Task Develops the concept of inverse functions in a linear modeling context using tables, graphs, and equations. (F.BF.1, F.BF.4, F.BF.4a) Ready, Set, Go Homework: Functions and Their Inverses 1.1

1.2Flipping Ferraris – A Solidify Understanding Task Extends the concepts of inverse functions in a quadratic modeling context with a focus on domain and range and whether a function is invertible in a given domain. (F.BF.1, F.BF.4, F.BF.4c, F.BF.4d)Ready, Set, Go Homework: Functions and Their Inverses 1.2

1.3Tracking the Tortoise – A Solidify Understanding Task Solidifies the concepts of inverse function in an exponential modeling context and surfaces ideas about logarithms. (F.BF.1, F.BF.4, F.BF.4c, F.BF.4d)Ready, Set, Go Homework: Functions and Their Inverses 1.3

1.4 Pulling a Rabbit Out of a Hat – A Solidify Understanding Task Uses function machines to model functions and their inverses. Focus on finding inverse functions and verifying that two functions are inverses. (F.BF.4, F.BF.4a, F.BF.4b) Ready, Set, Go Homework: Functions and Their Inverses 1.4

1.5 Inverse Universe – A Practice Understanding Task Uses tables, graphs, equations, and written descriptions of functions to match functions and their inverses together and to verify the inverse relationship between two functions. (F.BF.4a, F.BF.4b, F.BF.4c, F.BF.4d)Ready, Set, Go Homework: Functions and Their Inverses 1.5


FUNCTIONS AND THEIR INVERSES – 1.1


1.1 Brutus Bites Back

A Develop Understanding Task

RememberCarlosandClarita?Acoupleofyearsago,theystartedearningmoneybytakingcareofpetswhiletheirownersareaway.Duetotheiramazingmathematicalanalysisandtheirlovingcareofthecatsanddogsthattheytakein,CarlosandClaritahavemadetheirbusinessverysuccessful.Tokeepthehungrydogsfed,theymustregularlybuyBrutusBites,thefavoritefoodofallthedogs.

CarlosandClaritahavebeensearchingforanewdogfoodsupplierandhaveidentifiedtwopossibilities.TheCanineCateringCompany,locatedintheirtown,sells7poundsoffoodfor$5.

Carlosthoughtabouthowmuchtheywouldpayforagivenamountoffoodanddrewthisgraph:

1. WritetheequationofthefunctionthatCarlosgraphed.

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Claritathoughtabouthowmuchfoodtheycouldbuyforagivenamountofmoneyanddrewthisgraph:

2.WritetheequationofthefunctionthatClaritagraphed.

3. WriteaquestionthatwouldbemosteasilyansweredbyCarlos’graph.WriteaquestionthatwouldbemosteasilyansweredbyClarita’sgraph.Whatisthedifferencebetweenthetwoquestions?

4. Whatistherelationshipbetweenthetwofunctions?Howdoyouknow?

5.Usefunctionnotationtowritetherelationshipbetweenthefunctions.

2




Lookingonline,Carlosfoundacompanythatwillsell8poundsofBrutusBitesfor$6plusaflat$5shippingchargeforeachorder.Thecompanyadvertisesthattheywillsellanyamountoffoodatthesamepriceperpound.

6. ModeltherelationshipbetweenthepriceandtheamountoffoodusingCarlos’approach.

7. ModeltherelationshipbetweenthepriceandtheamountoffoodusingClarita’sapproach.

8. Whatistherelationshipbetweenthesetwofunctions?Howdoyouknow?

9. Usefunctionnotationtowritetherelationshipbetweenthefunctions.

10. WhichcompanyshouldClaritaandCarlosbuytheirBrutusBitesfrom?Why?

3


FUNCTIONS AND INVERSES – 1.1 1.1


Needhelp?Visitwww.rsgsupport.org

READYTopic:Inverseoperations

Inverseoperations“undo”eachother.Forinstance,additionandsubtractionareinverseoperations.

Soaremultiplicationanddivision.Inmathematics,itisoftenconvenienttoundoseveraloperations

inordertosolveforavariable.

Solveforxinthefollowingproblems.Thencompletethestatementbyidentifyingthe

operationyouusedto“undo”theequation.

1.24=3x Undomultiplicationby3by_______________________________________________

2. Undodivisionby5by______________________________________________________

3. Undoadd17by_____________________________________________________________

4. Undothesquarerootby___________________________________________________

5. Undothecuberootby_____________________________then___________________

6. Undoraisingxtothe4thpowerby________________________________________

7. Undosquaringby_______________________________then______________________

SET Topic:Linearfunctionsandtheirinverses

CarlosandClaritahaveapetsittingbusiness.Whentheyweretryingtodecidehowmanyeachof

dogsandcatstheycouldfitintotheiryard,theymadeatablebasedonthefollowinginformation.

Catpensrequire6ft2ofspace,whiledogrunsrequire24ft2.CarlosandClaritahaveupto360ft2

availableinthestorageshedforpensandruns,whilestillleavingenoughroomtomovearoundthe

cages.Theymadeatableofallofthecombinationsofcatsanddogstheycouldusetofillthespace.

Theyquicklyrealizedthattheycouldfitin4catsinthesamespaceasonedog.

x5= −2

x +17 = 20

x = 6

x +1( )3 = 2

x4 = 81

x − 9( )2 = 49

READY, SET, GO! Name PeriodDate

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FUNCTIONS AND INVERSES – 1.1

1.1



cats 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

dogs 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

8.Usetheinformationinthetabletowrite5orderedpairsthathavecatsastheinputvalue

anddogsastheoutputvalue.

9.Writeanexplicitequationthatshowshowmanydogstheycanaccommodatebasedonhow

manycatstheyhave.(Thenumberofdogs“d”willbeafunctionofthenumberofcats“c”or

! = #(%).)

10.Usetheinformationinthetabletowrite5orderedpairsthathavedogsastheinputvalue

andcatsastheoutputvalue.

11.Writeanexplicitequationthatshowshowmanycatstheycanaccommodatebasedonhow

manydogstheyhave.(Thenumberofcats“c”willbeafunctionofthenumberofdogs“d”

or% = '(!).)

Baseyouranswersin#12and#13onthetableatthetopofthepage.

12.Lookbackatproblem8andproblem10.Describehowtheorderedpairsaredifferent.

13.a)Lookbackattheequationyouwroteinproblem9.Describethedomainfor! = #(%).

b)Describethedomainfortheequation% = '(!)thatyouwroteinproblem11.

c)Whatistherelationshipbetweenthem?

5



1.1



GO Topic:Usingfunctionnotationtoevaluateafunction.

Thefunctions aredefinedbelow.

Calculatetheindicatedfunctionvaluesinthefollowingproblems.Simplifyyouranswers.

14.

15. 16. 17.

18.

19. 20. 20.)(* + ,)

22.

23. 24. 25.

f x( ), g x( ), and h x( )f x( ) = x g x( ) = 5x −12 h x( ) = x2 + 4x − 7

f 10( ) f −2( ) f a( ) f a + b( )

g 10( ) g −2( ) g a( )

h 10( ) h −2( ) h a( ) h a + b( )

6




1.2 Flipping Ferraris

A Solidify Understanding Task

Whenpeoplefirstlearntodrive,theyareoftentoldthatthefastertheyaredriving,thelongeritwilltaketostop.So,whenyou’redrivingonthefreeway,youshouldleavemorespacebetweenyourcarandthecarinfrontofyouthanwhenyouaredrivingslowlythroughaneighborhood.Haveyoueverwonderedabouttherelationshipbetweenhowfastyouaredrivingandhowfaryoutravelbeforeyoustop,afterhittingthebrakes?

1. Thinkaboutitforaminute.Whatfactorsdoyouthinkmightmakeadifferenceinhowfaracartravelsafterhittingthebrakes?

Therehasactuallybeenquiteabitofexperimentalworkdone(mostlybypolicedepartmentsandinsurancecompanies)tobeabletomathematicallymodeltherelationshipbetweenthespeedofacarandthebrakingdistance(howfarthecargoesuntilitstopsafterthedriverhitsthebrakes).

2. Imagineyourdreamcar.MaybeitisaFerrari550Maranello,asuper-fastItaliancar.Experimentshaveshownthatonsmooth,dryroads,therelationshipbetweenthebrakingdistance(d)andspeed(s)isgivenby!(#) = 0.03#) .Speedisgiveninmiles/hourandthedistanceisinfeet.a) Howmanyfeetshouldyouleavebetweenyouandthecarinfrontofyouifyouare

drivingtheFerrariat55mi/hr?

b) Whatdistanceshouldyoukeepbetweenyouandthecarinfrontofyouifyouaredrivingat100mi/hr?

c) Ifanaveragecarisabout16feetlong,abouthowmanycarlengthsshouldyouhavebetweenyouandthatcarinfrontofyouifyouaredriving100mi/hr?

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d) Itmakessensetoalotofpeoplethatifthecarismovingatsomespeedandthengoestwiceasfast,thebrakingdistancewillbetwiceasfar.Isthattrue?Explainwhyorwhynot.

3.Graphtherelationshipbetweenbrakingdistanced(s),andspeed(s),below.

4.AccordingtotheFerrariCompany,themaximumspeedofthecarisabout217mph.UsethistodescribeallthemathematicalfeaturesoftherelationshipbetweenbrakingdistanceandspeedfortheFerrarimodeledby!(#) = 0.03#) .

5.WhatifthedriveroftheFerrari550wascruisingalongandsuddenlyhitthebrakestostopbecauseshesawacatintheroad?Sheskiddedtoastop,andfortunately,missedthecat.Whenshegotoutofthecarshemeasuredtheskidmarksleftbythecarsothatsheknewthatherbrakingdistancewas31ft.

a)Howfastwasshegoingwhenshehitthebrakes?

b)Ifshedidn’tseethecatuntilshewas15feetaway,whatisthefastestspeedshecouldbetravelingbeforeshehitthebrakesifshewantstoavoidhittingthecat?

8




6.Partofthejobofpoliceofficersistoinvestigatetrafficaccidentstodeterminewhatcausedtheaccidentandwhichdriverwasatfault.Theymeasurethebrakingdistanceusingskidmarksandcalculatespeedsusingthemathematicalrelationshipsjustlikewehavehere,althoughtheyoftenusedifferentformulastoaccountforvariousfactorssuchasroadconditions.Let’sgobacktotheFerrarionasmooth,dryroadsinceweknowtherelationship.Createatablethatshowsthespeedthecarwastravelingbaseduponthebrakingdistance.

7.Writeanequationofthefunctions(d)thatgivesthespeedthecarwastravelingforagivenbrakingdistance.

8.Graphthefunctions(d)anddescribeitsfeatures.

9.Whatdoyounoticeaboutthegraphofs(d)comparedtothegraphofd(s)?Whatistherelationshipbetweenthefunctionsd(s)ands(d)?

9




10.Considerthefunction!(#) = 0.03#)overthedomainofallrealnumbers,notjustthedomainofthisproblemsituation.Howdoesthegraphchangefromthegraphofd(s)inquestion#3?

11.Howdoeschangingthedomainofd(s)changethegraphoftheinverseofd(s)?

12.Istheinverseofd(s)afunction?Justifyyouranswer.

10



1.2



READY Topic:SolvingforavariableSolveforx.

1. 17 = 5% + 2

2.2%( − 5 = 3%( − 12% +31

3.11 = √2% + 1

4.√%( + % − 2 = 25.−4 = √5% + 1- 6.√352- = √7%( + 9-

7.30 = 243 8.50 = 11(2 9.40 = 1

3(

SET Topic:Exploringinversefunctions

10.Studentsweregivenasetofdatatograph.Aftertheyhadcompletedtheirgraphs,each

studentsharedhisgraphwithhisshoulderpartner.WhenEthanandEmmasaweach

other’sgraphs,theyexclaimedtogether,“Yourgraphiswrong!”Neithergraphiswrong.

ExplainwhatEthanandEmmahavedonewiththeirdata.

Ethan’sgraph

Emma’sgraph


11



1.2



11.DescribeasequenceoftransformationsthatwouldtakeEthan’sgraphontoEmma’s.

12.Abaseballishitupwardfromaheightof3feetwith

aninitialvelocityof80feetpersecond(about55mph).Thegraphshowstheheightoftheballatanygivensecondduringitsflight.Usethegraphtoanswerthequestionsbelow.

a. Approximatethetimethattheballisatitsmaximumheight.

b. Approximatethetimethattheballhitstheground.

c. Atwhattimeistheball67feetabovetheground?

d. Makeanewgraphthatshowsthetimewhentheballisatthegivenheights.

e. Isyournewgraphafunction?Explain.

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1.2



GO Topic:Usingfunctionnotationtoevaluateafunction

Thefunctions aredefinedbelow.

Calculatetheindicatedfunctionvalues.Simplifyyouranswers.

13.4(7) 14.4(−9) 15.4(7) 16.4(7 − 8)

17.9(7) 18.9(−9) 19.9(7) 20.9(7 − 8)

21.ℎ(7) 22.ℎ(−9) 23.ℎ(7) 24.ℎ(7 − 8)

Noticethatthenotationf(g(x))isindicatingthatyoureplacexinf(x)withg(x).

Simplifythefollowing.

25.f(g(x)) 26.f(h(x)) 27.g(f(x))

f x( ), g x( ), and h x( )

f x( ) = 3x g x( ) = 10x + 4 h x( ) = x2 − x

13




1.3 Tracking the Tortoise

A Solidify Understanding Task

Youmayrememberataskfromlastyearaboutthefamousracebetweenthetortoiseandthehare.Inthechildren’sstoryofthetortoiseandthehare,theharemocksthetortoiseforbeingslow.Thetortoisereplies,“Slowandsteadywinstherace.”Theharesays,“We’lljustseeaboutthat,”andchallengesthetortoisetoarace.

Inthetask,wemodeledthedistancefromthestartinglinethatboththetortoiseandtheharetravelledduringtherace.Todaywewillconsideronlythejourneyofthetortoiseintherace.

Becausethehareissoconfidentthathecanbeatthetortoise,hegivesthetortoisea1meterheadstart.Thedistancefromthestartinglineofthetortoiseincludingtheheadstartisgivenbythefunction:

!(#) = 2( (dinmetersandtinseconds)

Thetortoisefamilydecidestowatchtheracefromthesidelinessothattheycanseetheirdarlingtortoisesister,Shellie,provethevalueofpersistence.

1. Howfarawayfromthestartinglinemustthefamilybe,tobelocatedintherightplaceforShellietorunby5secondsafterthebeginningoftherace?After10seconds?

2. Describethegraphofd(t),Shellie’sdistanceattimet.Whataretheimportantfeaturesofd(t)?

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3. Ifthetortoisefamilyplanstowatchtheraceat64metersawayfromShellie’sstartingpoint,howlongwilltheyhavetowaittoseeShellierunpast?

4. HowlongmusttheywaittoseeShellierunbyiftheystand1024metersawayfromher

startingpoint?

5. DrawagraphthatshowshowlongthetortoisefamilywillwaittoseeShellierunbyatagivenlocationfromherstartingpoint.

6. HowlongmustthefamilywaittoseeShellierunbyiftheystand220metersawayfrom

herstartingpoint?

7. Whatistherelationshipbetweend(t)andthegraphthatyouhavejustdrawn?Howdidyouused(t)todrawthegraphin#5?

15




8. Considerthefunction)(*) = 2+ .A) Whatarethedomainandrangeof)(*)?Is)(*)invertible?

B) Graph)(*)and)01(*)onthegridbelow.

C) Whatarethedomainandrangeof)01(*)?

9. If)(3) = 8,whatis)01(8)?Howdoyouknow?

10. If) 5167 = 1.414,whatis)01(1.414)?Howdoyouknow?

11. If)(;) = <whatis)01(<)?Willyouranswerchangeiff(x)isadifferentfunction?Explain.

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1.3



READY Topic:Solvingexponentialequations.Solveforthevalueofx.

1. 5"#$ = 5&"'( 2. 7("'& = 7'&"#* 3. 4(" = 2&"'*

4. 3."'/ = 9&"'( 5. 8"#$ = 2&"#( 6. 3"#$ = $

*$

SET Topic:ExploringtheinverseofanexponentialfunctionInthefairytaleJackandtheBeanstalk,Jackplantsamagicbeanbeforehegoestobed.InthemorningJackdiscoversagiantbeanstalkthathasgrownsolarge,itdisappearsintotheclouds.Buthereisthepartofthestoryyouneverheard.Writtenonthebagcontainingthemagicbeanswasthisnote.Plant a magic bean in rich soil just as the sun is setting. Do not look at the plant site for 10 hours. (This is part of the magic.) After the bean has been in the ground for 1 hour, the growth of the sprout can be modeled by the function 2(4) = 36. (b in feet and t in hours) Jackwasagoodmathstudent,soalthoughheneverlookedathisbeanstalkduringthenight,heusedthefunctiontocalculatehowtallitshouldbeasitgrew.Thetableontherightshowsthecalculationshemadeeveryhalfhour.Hence,Jackwasnotsurprisedwhen,inthemorning,hesawthatthetopofthebeanstalkhaddisappearedintotheclouds.

Time(hours) Height(feet)1 31.5 5.22 92.5 15.63 273.5 46.84 814.5 140.35 2435.5 420.96 7296.5 1,262.77 2,1877.5 3,7888 6,5618.5 11,3649 19,6839.5 34,09210 59,049


17



1.3



7.DemonstratehowJackusedthemodel2(4) = 36 tocalculatehowhighthebeanstalkwouldbe

after6hourshadpassed.(Youmayusethetablebutwritedownwhereyouwouldputthenumbersinthefunctionifyoudidn’thavethetable.)

8.Duringthatsamenight,aneighborwasplayingwithhisdrone.Itwasprogrammedtohoverat

243ft.Howmanyhourshadthebeanstalkbeengrowingwhenitwasashighasthedrone?9.Didyouusethetableinthesamewaytoanswer#8asyoudidtoanswer#7? Explain.10.WhileJackwasmakinghistable,hewaswonderinghowtallthebeanstalkwouldbeafterthe

magical10hourshadpassed.Hequicklytypedthefunctionintohiscalculatortofindout.WritetheequationJackwouldhavetypedintohiscalculator.

11.Commercialjetsflybetween30,000ft.and36,000ft.Abouthowmanyhoursofgrowingcould

passbeforethebeanstalkmightinterferewithcommercialaircraft?Explainhowyougotyouranswer.

12.Usethetabletofind9(7)and9'$(11,364).13.Usethetabletofind9(9)and9'$(9).13.Explainwhyit’spossibletoanswersomeofthequestionsabouttheheightofthebeanstalkby

justpluggingthenumbersintothefunctionruleandwhysometimesyoucanonlyusethetable.

18



1.3



GO Topic:EvaluatingfunctionsThefunctions aredefinedbelow.

f(x) = −2x g(x) = 2x + 5 h(x) = x& + 3x − 10

Calculatetheindicatedfunctionvalues.Simplifyyouranswers.

14.f(a)

15.f(b&) 16.f(a + b) 17.fFG(H)I

18.g(a)

19.g(b&) 20.g

(a + b) 21.hFf(H)I

22.h(a)

23.h(b&) 24.h

(a + b) 25.hFG(H)I

f x( ), g x( ), and h x( )

19




1.4 Pulling a Rabbit Out of the Hat A Solidify Understanding Task Ihaveamagictrickforyou:

• Pickanumber,anynumber.• Add6• Multiplytheresultby2• Subtract12• Divideby2• Theansweristhenumberyoustartedwith!

Peopleareoftenmystifiedbysuchtricksbutthoseofuswhohavestudiedinverseoperationsandinversefunctionscaneasilyfigureouthowtheyworkandevencreateourownnumbertricks.Let’sgetstartedbyfiguringouthowinversefunctionsworktogether.

Foreachofthefollowingfunctionmachines,decidewhatfunctioncanbeusedtomaketheoutputthesameastheinputnumber.Describetheoperationinwordsandthenwriteitsymbolically.

Here’sanexample:

Input Output

!(#) = # + 8 !)*(#) = # − 8

# = 7 7 7 + 8 = 15

Inwords:Subtract8fromtheresult

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1.

2.

Inwords:

Input Output

!(#) = 3# !)*(#) =

# = 7 7 3 ∙ 7 = 21

Input Output

!(#) = #/ !)*(#) =

# = 7 7 7/ = 49

Inwords:

21




3.

4.

5.

Inwords:

Input Output

!(#) = 23 !)*(#) =

# = 7 7 24 = 128

Input Output

!(#) = 2# − 5 !)*(#) =

# = 7 7 2 ∙ 7 − 5 = 9

Input Output

!(#) = # + 53 !)*(#) =

# = 7 7 7 + 53 = 4

Inwords:

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6.

7.

Input Output

!(#) = (# − 3)/ !)*(#) =

# = 7 7 (7 − 3)/ = 16

Input Output

!(#) = 4 − √# !)*(#) =

# = 7 7 4 − √7

Inwords:

Inwords:

Inwords:

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8.

9.Eachoftheseproblemsbeganwithx=7.Whatis thedifferencebetweenthe#usedin!(#)andthe#used in!)*(#)?

10.In#6,couldanyvalueof#beusedin!(#)andstillgivethesameoutputfrom!)*(#)?Explain.Whatabout#7?

11.Basedonyourworkinthistaskandtheothertasksinthismodulewhatrelationshipsdoyouseebetweenfunctionsandtheirinverses?

Inwords:

Input Output

!(#) = 23 − 10 !)*(#) =

# = 7 7 24 − 10 = 118

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1.4



READY Topic:PropertiesofexponentsUsetheproductruleorthequotientruletosimplify.Leaveallanswersinexponentialformwithonlypositiveexponents.

1. 3" ∙ 3$

2.7& ∙ 7" 3.10)* ∙ 10+ 4. 5- ∙ 5)"

5. .&.$

6. 2" ∙ 2)0 ∙ 2 7. 1221)$ 8. +3

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10.0305 11.

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SET Topic:Inversefunction13. Giventhefunctions:(<) = √< − 1BCDE(<) = <& + 7:

a.Calculate:(16)BCDE(3).

b.Write:(16)asanorderedpair.

c.WriteE(3)asanorderedpair.

d.Whatdoyourorderedpairsfor:(16)andE(3)imply?

e.Find:(25).

f.Basedonyouranswerfor:(25),predictE(4).

g.FindE(4). Didyouranswermatchyourprediction?

h.Are:(<)BCDE(<)inversefunctions? Justifyyouranswer.


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1.4



Matchthefunctioninthefirstcolumnwithitsinverseinthesecondcolumn.

:(<) :)2(<)16.:(<) = 3< + 5

a.:)2(<) = KLE$<

17.:(<) = <$ b.:)2(<) = √<9

18.:(<) = √< − 33 c.:)2(<) = M)$0

19.:(<) = <0 d.:)2(<) = M0 − 5

20.:(<) = 5M e.:)2(<) = KLE0<

21.:(<) = 3(< + 5) f.:)2(<) = <$ + 3

22.:(<) = 3M g.:)2(<) = √<3

GO Topic:Compositefunctionsandinverses

CalculateNOP(Q)RSTUPON(Q)Rforeachpairoffunctions.(Note:thenotation(: ∘ E)(<)BCD(E ∘ :)(<)meansthesamethingas:OE(<)RBCDEO:(<)R,respectively.)

23.:(<) = 2< + 5E(<) = M)$

&

24.:(<) = (< + 2)0E(<) = √<9 − 2

25.:(<) = 0* < + 6E(<) =

*(M)")0

26.:(<) = )0M + 2E(<) =

)0M)&

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1.4



Matchthepairsoffunctionsabove(23-26)withtheirgraphs.Labelf(x)andg(x).

a. b.

c. d.

27.Graphtheliney=xoneachofthegraphsabove.Whatdoyounotice?

28.Doyouthinkyourobservationsaboutthegraphsin#27hasanythingtodowiththe

answersyougotwhenyoufound:OE(<)RBCDEO:(<)R?Explain.

29.Lookatgraphb.Shadethe2trianglesmadebythey-axis,x-axis,andeachline.Whatis

interestingaboutthesetwotriangles?

30.Shadethe2trianglesingraphd.Aretheyinterestinginthesameway?Explain.

27




1.5 Inverse Universe A Practice Understanding Task Youandyourpartnerhaveeachbeengivenadifferent

setofcards.Theinstructionsare:

1. Selectacardandshowittoyourpartner.2. Worktogethertofindacardinyourpartner’sset

ofcardsthatrepresentstheinverseofthefunctionrepresentedonyourcard.

3. Recordthecardsyouselectedandthereasonthatyouknowthattheyareinversesinthespacebelow.

4. Repeattheprocessuntilallofthecardsarepairedup.

*Forthistaskonly,assumethatalltablesrepresentpointsonacontinuousfunction.

Pair1: _____________________ Justificationofinverserelationship:____________________________________





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Pair10:_____________________ Justificationofinverserelationship:____________________________________

29

A1

𝑓(𝑥) = {−2𝑥 − 2, −5 < 𝑥 < 0−2, 𝑥 ≥ 0

A3 Each input value, 𝑥, is squared and then 3 is added to the result. The domain of the function is [0,∞)

A5

x y -2 -3 2 3 0 0 6 5 4 4

−43

-2

A2 The function increases at a constant rate of 𝑎

𝑏 and the y-intercept is (0, c).

A4

A6

𝑦 = 3𝑥

A8

A7

x y -5 -125 -3 -27 -1 -1 1 1 3 27 5 125

A9

A10 Yasmin started a savings account with $5. At the end of each week, she added 3. This function models the amount of money in the account for a given week.

B1

𝑦 = log3 𝑥

B2

𝑓(𝑥) = { 23 𝑥, −3 < 𝑥 < 3

2𝑥 − 4, 𝑥 ≥ 3

B4

x y -216 -6 -64 -4 -8 -2 0 0 8 2

64 4 216 6

B3 The x-intercept is (c, 0) and the slope of the line is 𝑏

𝑎.

B6

x y 3 0 4 1 7 2

12 3 19 4 28 5 39 6

B5

B7

B8

𝒙 y -2 -3 -1 -2 0 1 1 6 2 13

B9

B10 The function is continuous and grows by an equal factor of 5 over equal intervals. The y-intercept is (0,1).



1.5



READY Topic:PropertiesofexponentsUsepropertiesofexponentstosimplifythefollowing.Writeyouranswersinexponentialformwithpositiveexponents.1.√"#$ ∙ √"&$

2.√"' ∙ √"( ∙ √")

3.√*) ∙ √*#' ∙ √+&,

4.√32, ∙ √9 ∙ √27'

5.√8( ∙ √16' ∙ √2)

6.(5#)&

7.(7#)78

8.(379)7:

9.;:<(

:=>&

SET Topic:RepresentationsofinversefunctionsWritetheinverseofthegivenfunctioninthesameformatasthegivenfunction.Functionf(x)

Inverse?78(")

10.x f(x)

-8 0

-4 3

0 6

4 9

8 12

10.


30



1.5



11. 12.

12.?(") = −2" + 4

13.?(") = EFG&"

14.

15.x ?(")

0 0

1 1

2 4

3 9

4 16

15.

31



1.5



GO Topic:CompositefunctionsCalculateHIJ(K)LMNOJIH(K)Lforeachpairoffunctions.

(Note:thenotation(? ∘ G)(")*QR(G ∘ ?)(")meanthesamething,respectively.)

16.?(") = 3" + 7; G(") = −4" − 11

17.?(") = −4" + 60; G(") = − 89" + 15

18.?(") = 10" − 5; G(") = #: " + 3

19.?(") = −#& " + 4; G(") = − &

# " + 6

20.Lookbackatyourcalculationsfor?IG(")L*QRGI?(")L.Twoofthepairsofequationsare

inversesofeachother.Whichonesdoyouthinktheyare?

Why?

32

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