ALGEBRA II - Mathematics Vision ProjectUses tables, graphs, equations, and written descriptions of...

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

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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.

Claritathoughtabouthowmuchfoodtheycouldbuyforagivenamountofmoneyanddrewthisgraph:

2.WritetheequationofthefunctionthatClaritagraphed.

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

4. Whatistherelationshipbetweenthetwofunctions?Howdoyouknow?

5.Usefunctionnotationtowritetherelationshipbetweenthefunctions.

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

6. ModeltherelationshipbetweenthepriceandtheamountoffoodusingCarlos’approach.

7. ModeltherelationshipbetweenthepriceandtheamountoffoodusingClarita’sapproach.

8. Whatistherelationshipbetweenthesetwofunctions?Howdoyouknow?

9. Usefunctionnotationtowritetherelationshipbetweenthefunctions.

10. WhichcompanyshouldClaritaandCarlosbuytheirBrutusBitesfrom?Why?

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 +1( )3 = 2

x4 = 81

x − 9( )2 = 49

READY, SET, GO! Name PeriodDate

FUNCTIONS AND INVERSES – 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?

GO Topic:Usingfunctionnotationtoevaluateafunction.

Thefunctions aredefinedbelow.

Calculatetheindicatedfunctionvaluesinthefollowingproblems.Simplifyyouranswers.

15. 16. 17.

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

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( )

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?

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?

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)?

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

11.Howdoeschangingthedomainofd(s)changethegraphoftheinverseofd(s)?

12.Istheinverseofd(s)afunction?Justifyyouranswer.

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

SET Topic:Exploringinversefunctions

10.Studentsweregivenasetofdatatograph.Aftertheyhadcompletedtheirgraphs,each

studentsharedhisgraphwithhisshoulderpartner.WhenEthanandEmmasaweach

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

ExplainwhatEthanandEmmahavedonewiththeirdata.

Ethan’sgraph

Emma’sgraph

11.DescribeasequenceoftransformationsthatwouldtakeEthan’sgraphontoEmma’s.

12.Abaseballishitupwardfromaheightof3feetwith

aninitialvelocityof80feetpersecond(about55mph).Thegraphshowstheheightoftheballatanygivensecondduringitsflight.Usethegraphtoanswerthequestionsbelow.

a. Approximatethetimethattheballisatitsmaximumheight.

b. Approximatethetimethattheballhitstheground.

c. Atwhattimeistheball67feetabovetheground?

d. Makeanewgraphthatshowsthetimewhentheballisatthegivenheights.

e. Isyournewgraphafunction?Explain.

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

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)?

3. Ifthetortoisefamilyplanstowatchtheraceat64metersawayfromShellie’sstartingpoint,howlongwilltheyhavetowaittoseeShellierunpast?

4. HowlongmusttheywaittoseeShellierunbyiftheystand1024metersawayfromher

startingpoint?

5. DrawagraphthatshowshowlongthetortoisefamilywillwaittoseeShellierunbyatagivenlocationfromherstartingpoint.

6. HowlongmustthefamilywaittoseeShellierunbyiftheystand220metersawayfrom

herstartingpoint?

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

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.

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

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.

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( )

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

ristia

Inwords:

Input Output

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

# = 7 7 3 ∙ 7 = 21

Input Output

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

# = 7 7 7/ = 49

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:

Input Output

!(#) = (# − 3)/ !)*(#) =

# = 7 7 (7 − 3)/ = 16

Input Output

!(#) = 4 − √# !)*(#) =

# = 7 7 4 − √7

Inwords:

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

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

10.0305 11.

+67+65 12. 8

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.

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(<) =

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

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.

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:____________________________________

Pair10:_____________________ Justificationofinverserelationship:____________________________________

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,∞)

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

A2 The function increases at a constant rate of 𝑎

𝑏 and the y-intercept is (0, c).

𝑦 = 3𝑥

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

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.

𝑦 = log3 𝑥

𝑓(𝑥) = { 23 𝑥, −3 < 𝑥 < 3

2𝑥 − 4, 𝑥 ≥ 3

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

x y 3 0 4 1 7 2

12 3 19 4 28 5 39 6

𝒙 y -2 -3 -1 -2 0 1 1 6 2 13

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

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)

11. 12.

12.?(") = −2" + 4

13.?(") = EFG&"

15.x ?(")

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?

ALGEBRA II - Mathematics Vision ProjectUses tables, graphs, equations, and written descriptions of...

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Graphs of Functions and Inverses What is the connection between the graphs of functions and their inverses?

3-4: Inverse Functions and Relations - Welcome to Mrs. …plankmathclass.weebly.com/.../3-4-_inverse_functions_… · · 2017-10-30¥ Graph functions and their inverses. Two relations

* Objectives: Students will be able to… * Determine whether a function has an inverse * Write an inverse function * Verify 2 functions are inverses of

3.4 Inverse Functions Goal: Find and use inverses of linear and nonlinear functions

8-7 Radical · PDF file8-7 Radical Functions 619 Recall that exponential and logarithmic functions are inverse functions. Quadratic and cubic functions have inverses as well

Functions, properties. elementary functions and their inverses

Functions and Their Inverses - Humble Independent … · to verify that 2 functions are inverses. ... 9-5 Functions and Their Inverses Lesson Quiz: Part I A: yes; B: no 1. Use the

4-2 Inverse Functions. * Right Answer : D Graph and recognize inverses of relations and functions. Find inverses of functions

6.4 Inverse Functions Part 1 Goal: Find inverses of linear functions

11.6.14 Inverse Functions Wrap Up · 2014. 11. 6. · Inverse Functions Wrap Up Wrap Up: Notation for Inverse of !(!):___ Inverses switch the _ and the ___. Graphs of inverses

Unit 6. Exponential Functions and an Introduction to …...will extend their understanding of inverse functions as they explore logarithms as inverses to exponential functions and

Inverse Functions Section 7.4. WHAT YOU WILL LEARN: 1.How to find the inverses of linear functions. 2.How to find inverses of nonlinear functions

INVERSE FUNCTIONS. Prove that and are inverses of each other Complete warm up individually and then compare to a neighbor END IN MIND

+ Inverse Functions How do we verify and find inverses of functions? M2 Unit 5a: Day 3