Functions & Their Inverses - mathematicsvisionproject.org · SECONDARY MATH 3 // MODULE 1 FUNCTIONS...

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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MODULE 1

Functions & Their Inverses

SECONDARY

MATH THREE

An Integrated Approach

SECONDARY MATH 3 // 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

SECONDARY MATH III // MODULE 1

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

Needhelp?Visitwww.rsgsupport.org

READY Topic:InverseoperationsInverseoperations“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.

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

x5= −2

x +17 = 20

x = 6

x +1( )3 = 2

x4 = 81

x − 9( )2 = 49

READY, SET, GO! Name PeriodDate

4




1.1


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% + 313.11 = 2% + 1

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

7.9/ = 243 8.5/ = 00(1 9.4/ = 0

2(

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.

12




1.2


GO Topic:Usingfunctionnotationtoevaluateafunction

Thefunctions f x( ), g x( ), and h x( ) aredefinedbelow.

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

Calculatetheindicatedfunctionvalues.Simplifyyouranswers.

13.3 7 14.3 −9 15.3 4 16.3 4 − 5

17.6 7 18.6 −9 19.6 4 20.6 4 − 5

21.ℎ 7 22.ℎ −9 23.ℎ 4 24.ℎ 4 − 5

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

Simplifythefollowing.

25.f(g(x)) 26.f(h(x)) 27.g(f(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. HowlongmusttheywaittoseeShellierunbyiftheystand1024metersawayfromherstartingpoint?

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.

16




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 tocalculatehowhighthebeanstalkwouldbeafter6hourshadpassed.(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.

3.

Inwords:

Input Output

!(#) = 2. !)*(#) =

# = 7 7 2/ = 128

Inwords:

Input Output

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

# = 7 7 3 ∙ 7 = 21

Input Output

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

# = 7 7 73 = 49

Inwords:

21




4.

5.

6.

Input Output

!(#) = 2# − 5 !)*(#) =

# = 7 7 2 ∙ 7 − 5 = 9

Input Output

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

# = 7 7 7 + 53 = 4

Input Output

!(#) = (# − 3)3 !)*(#) =

# = 7 7 (7 − 3)3 = 16

Inwords:

Inwords:

Inwords:

22




7.

8.

9.Eachoftheseproblemsbeganwithx=7.Whatisthedifferencebetweenthe#usedin!(#)andthe#usedin!)*(#)?

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

11.Basedonyourworkinthistaskandtheothertasksinthismodulewhatrelationshipsdoyouseebetweenfunctionsandtheirinverses?

Input Output

!(#) = 4 − √# !)*(#) =

# = 7 7 4 − √7

Inwords:

Inwords:

Input Output

!(#) = 2. − 10 !)*(#) =

# = 7 7 2/ − 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

+4

9.-5

-

10.03

05 11.

+67

+65 12.8

69

83

SET Topic:Inversefunction13. Giventhefunctions: ; = ; − 1?@AB ; = ;& + 7:

a.Calculate: 16 ?@AB 3 .

b.Write: 16 asanorderedpair.

c.WriteB 3 asanorderedpair.

d.Whatdoyourorderedpairsfor: 16 andB 3 imply?

e.Find: 25 .

f.Basedonyouranswerfor: 25 ,predictB 4 .

g.FindB 4 . Didyouranswermatchyourprediction?

h.Are: ; ?@AB ; inversefunctions? Justifyyouranswer.


24




1.4


Matchthefunctioninthefirstcolumnwithitsinverseinthesecondcolumn.

: ; :)2 ; 16.: ; = 3; + 5

a.:)2 ; = HIB$;

17.: ; = ;$ b.:)2 ; = ;9

18.: ; = ; − 33 c.:)2 ; =J)$

0

19.: ; = ;0 d.:)2 ; =J

0− 5

20.: ; = 5J e.:)2 ; = HIB0;

21.: ; = 3 ; + 5 f.:)2 ; = ;$ + 3

22.: ; = 3J g.:)2 ; = ;3

GO Topic:Compositefunctionsandinverses

CalculateK L M NOPL K M foreachpairoffunctions.

(Note:thenotation : ∘ B ; ?@A B ∘ : ; meansthesamethingas: B ; ?@AB : ; ,

respectively.)

23.: ; = 2; + 5B ; =J)$

&

24.: ; = ; + 2 0B ; = ;9 − 2

25.: ; =0

*; + 6B ; =

* J)"

0

26.: ; =)0

J+ 2B ; =

)0

J)&

25




1.4


Match the pairs of functions above (23-26) with their graphs. Label f (x) and g (x). a. b.

c. d.

27.Graphtheliney=xoneachofthegraphsabove.Whatdoyounotice?

28.Doyouthinkyourobservationsaboutthegraphsin#27hasanythingtodowiththe

answersyougotwhenyoufound: B ; ?@AB : ; ?Explain.

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

interestingaboutthesetwotriangles?

30.Shadethe2trianglesingraphd.Aretheyinterestinginthesameway?Explain.

26



© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org

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:____________________________________

28




A1

!(#) = &−2# − 2, −5 < # < 0−2, # ≥ 0

A3

Eachinputvalue,#,issquaredandthen3isaddedtotheresult.Thedomainofthe

functionis[0,∞)

A5

x y-2 -3

2 3

0 0

6 5

4 4

−43

-2

A2

Thefunctionincreasesataconstant

rateof01andthey-interceptis(0,c).

A4

A6

2 = 33

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A8

A7

x y-5 -125

-3 -27

-1 -1

1 1

3 27

5 125

A9

A10

Yasminstartedasavingsaccountwith

$5.Attheendofeachweek,sheadded

3.Thisfunctionmodelstheamountof

moneyintheaccountforagivenweek.

30




B1

2 = log7#

B2

!(#) = 923 #, −3 < # < 32# − 4, # ≥ 3

B4

x y-216 -6

-64 -4

-8 -2

0 0

8 2

64 4

216 6

B3

Thex-interceptis(c,0)andtheslope

ofthelineis10.

B6

x y3 0

4 1

7 2

12 3

19 4

28 5

39 6

B5

31




B7

B8

: y-2 -3

-1 -2

0 1

1 6

2 13

B9

B10

Thefunctioniscontinuousandgrows

byanequalfactorof5overequal

intervals.They-interceptis(0,1).

32




1.5


READY Topic:PropertiesofexponentsUsepropertiesofexponentstosimplifythefollowing.Writeyouranswersinexponentialformwithpositiveexponents.1. !"

#∙ !%#

2. !& ∙ !' ∙ !(

3. )( ∙ )"

&∙ *%+

4. 32+ ∙ 9 ∙ 27&

5. 8' ∙ 16& ∙ 2(

6. 5" %

7. 7" 45

8. 346 47

9. 78'

79

%

SET Topic:RepresentationsofinversefunctionsWritetheinverseofthegivenfunctioninthesameformatasthegivenfunction.Functionf(x)

Inverse:45 !

10.x f(x)

-8 0

-4 3

0 6

4 9

8 12

10.


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1.5


11. 12.

12.:(!) = −2! + 4

13.: ! = BCD%!

14.

15.x : !

0 0

1 1

2 4

3 9

4 16

15.

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1.5


GO Topic:CompositefunctionsCalculateE F G HIJF E G foreachpairoffunctions.

(Note:thenotation : ∘ D ! )LM D ∘ : ! meanthesamething,respectively.)

16.: ! = 3! + 7; D ! = −4! − 11

17.: ! = −4! + 60; D ! = −5

6! + 15

18.: ! = 10! − 5; D ! ="

7! + 3

19.: ! = −"

%! + 4; D ! = −

%

"! + 6

20.Lookbackatyourcalculationsfor: D ! )LMD : ! .Twoofthepairsofequationsare

inversesofeachother.Whichonesdoyouthinktheyare?

Why?

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Functions & Their Inverses - mathematicsvisionproject.org · SECONDARY MATH 3 // MODULE 1 FUNCTIONS...

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