Functions and Their Inverses - Mathematics Vision Project · 2020-02-06 · ALGEBRA II // MODULE 1...

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

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 1

Functions and Their Inverses

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.

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

6. ModeltherelationshipbetweenthepriceandtheamountoffoodusingCarlos’approach.

7. ModeltherelationshipbetweenthepriceandtheamountoffoodusingClarita’sapproach.

8. Whatistherelationshipbetweenthesetwofunctions?Howdoyouknow?

9. Usefunctionnotationtowritetherelationshipbetweenthefunctions.

10. WhichcompanyshouldClaritaandCarlosbuytheirBrutusBitesfrom?Why?

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1.1 Brutus Bites Back – Teacher Notes A Develop Understanding Task

Purpose:Thepurposeofthistaskistodevelopandextendtheconceptofinversefunctionsina

linearcontext.Inthetask,studentsusetables,graphs,andequationstorepresentinversefunctions

astwodifferentwaysofmodelingthesamesituation.Therepresentationsexposetheideathatthe

domainofthefunctionistherangeoftheinverse(andviceversa)forsuitablyrestricteddomains.

Studentsmayalsonoticethatthegraphsoftheinversefunctionsarereflectionsoverthe𝑦 = 𝑥

line,butwiththeunderstandingthattheaxesarepartofthereflection.

CoreStandardsFocus:

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F.BF.4Findinversefunctions.

a. Solveanequationoftheform𝑓(𝑥) = 𝑐forasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.

RelatedStandards:

F-BF.4.b,F.BF.4c

StandardsforMathematicalPractice:

SMP1–Makesenseofproblemsandpersevereinsolvingthem

SMP7–Lookforandmakeuseofstructure

Vocabulary:Inversefunction

TheTeachingCycle:

Launch(WholeClass):Beginthetaskbyensuringthatstudentsunderstandthecontext.Ifthey

havenotdonethe“PetSitters”unitfromAlgebraI,theymaynotbefamiliarwiththestoryofCarlos

andClarita,twostudentsthatstartedabusinesscaringforpets.Theimportantpartofthecontext

issimplythattheyareplanningtobuydogfoodthatispricedbythepound.Askstudentstobegin

byworkingindividuallyonquestions1,2,and3.Circulatearoundtheroomlookingathow




studentsareworkingwiththegraphs.Identifystudentstopresenttotheclassthathave

constructedatableofvaluestoaidinwritingtheequation.Alsowatchtoseethevariablesthat

studentsareusingastheywriteequations.Iftheyareusing𝑥astheindependentvariableforboth

equations,thentheyhavenotproperlydefinedthevariable.Ifthisisacommonerror,youmay

choosetodiscussitwiththeclass.

Oncestudentshavewrittentheirequations,askastudenttopresentatablefor#1.Identifythe

differencebetweentermsinthetableandthenhaveastudentpresentanequationthatuses

variableslikep(pounds)andd(dollars)andwritesanequationfordollarsasafunctionofpounds.

Thenhaveastudentpresentatableofvaluesfor#2.Asktheclasstocomparethetablefor#1to

thetablefor#2.Theyshouldnoticethatthepanddcolumnhavebeenswitched,makingdthe

independentvariableandpthedependentvariable.Theyshouldalsonoticethechangeinthe

differencebetweentermsonthetwotables.Askforastudenttopresenttheequationwrittenfor

#2.Then,discussquestion#3.BesurethatstudentsunderstandthatCarlosthinksabouthow

muchitwillcostforacertainnumberofpounds,whileClaritathinksabouthowmuchfoodshecan

buywithacertainamountofmoney.

Asktheclasstorespondtoquestion#4.Theyshouldhavesomeexperiencewithinversefunctions

fromSecondaryMathematicsII,sotheyshouldrecognizetheminthiscontext.Modelthenotation

for#5as:𝐷(𝑝) = 𝑃,-𝑎𝑛𝑑𝑃(𝑑) =𝐷,-.Askstudentsforthedomainandrangeofeachfunction.

Explore(SmallGroup):Atthispoint,letstudentsworktogetherontheremainingproblems.As

youmonitorstudentsastheywork,encouragetheuseoftablestosupporttheirwritingof

equations.

Discuss(WholeClass):Proceedwiththediscussionmuchlikethepreviousdiscussionduringthe

launch.Presentthetable,graph,andequationfor#6andthenthesameprocessfor#7.Besure

thatstudentscanrelatetheirequationsbacktothecontext.Forinstance,besurethattheycan

identifywhythe5isaddedorsubtractedintheequations.Helpstudentstoseehowthestructure

oftheequationofthefunctionisreversedintheinversefunctiononestepatatime.Tiethiswork

tothestorycontexttogiveitmeaning.Again,askstudentstocomparethedomainandrangefor

thefunctions.Besurethattheynoticethatthedomainofthefunctionistherangeoftheinverse

andviceversa.




Note:Problem#10isanextensionproblemforstudentsthatmaybeaheadoftheclassin

completingthetask.Itisintendedtogivestudentsanopportunityforadditionalanalysis,

butisnotcriticalforthedevelopmentoftheideasofinverseinthismodule.

AlignedReady,Set,Go:FunctionsandTheirInverses1.1


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

𝑑 = 𝑓(𝑐).)

10.Usetheinformationinthetabletowrite5orderedpairsthathavedogsastheinputvalue

andcatsastheoutputvalue.

11.Writeanexplicitequationthatshowshowmanycatstheycanaccommodatebasedonhow

manydogstheyhave.(Thenumberofcats“c”willbeafunctionofthenumberofdogs“d”

or𝑐 = 𝑔(𝑑).)

Baseyouranswersin#12and#13onthetableatthetopofthepage.

12.Lookbackatproblem8andproblem10.Describehowtheorderedpairsaredifferent.

13.a)Lookbackattheequationyouwroteinproblem9.Describethedomainfor𝑑 = 𝑓(𝑐).

b)Describethedomainfortheequation𝑐 = 𝑔(𝑑)thatyouwroteinproblem11.

c)Whatistherelationshipbetweenthem?

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1.1



GO Topic:Usingfunctionnotationtoevaluateafunction.

Thefunctions aredefinedbelow.

Calculatetheindicatedfunctionvaluesinthefollowingproblems.Simplifyyouranswers.

14.

15. 16. 17.

18.

19. 20. 20.𝒈(𝒂 + 𝒃)

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( )

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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)isgivenby𝑑(𝑠) = 0.03𝑠) .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.UsethistodescribeallthemathematicalfeaturesoftherelationshipbetweenbrakingdistanceandspeedfortheFerrarimodeledby𝑑(𝑠) = 0.03𝑠) .

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

a)Howfastwasshegoingwhenshehitthebrakes?

b)Ifshedidn’tseethecatuntilshewas15feetaway,whatisthefastestspeedshecouldbetravelingbeforeshehitthebrakesifshewantstoavoidhittingthecat?

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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)?

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

11.Howdoeschangingthedomainofd(s)changethegraphoftheinverseofd(s)?

12.Istheinverseofd(s)afunction?Justifyyouranswer.

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1.2 Flipping Ferraris – Teacher Notes A Solidify Understanding Task

SpecialNotetoTeachers:Studentsmaybeinterestedtoknowthattheinformationaboutthe

Ferrariprovidedinthistaskisbaseduponactualstudiesperformedtotestthecarsatvarious

speeds.Theunitsforspeedanddistancewerechangedfrommetrictothemorefamiliarimperial

unitssothatstudentscouldusetheirintuitiveknowledgetothinkaboutthereasonablenessoftheir

answers.

Purpose:Thepurposeofthistaskistoextendstudents’understandingofinversefunctionsto

includequadraticfunctionsandsquareroots.Inthetask,studentsaregiventheequationofa

quadraticfunctionforbrakingdistanceandaskedtothinkaboutthedistanceneededtostopsafely

foragivenspeed.Thelogicisthenturnedaroundandstudentsareaskedtomodelthespeedthe

carwasgoingforagivenbrakingdistance.Afterstudentshaveworkedwiththeinverse

relationshipswiththelimiteddomain[0,217](assumingthat217isthemaximumspeedofthe

car),thenstudentsareaskedtoconsiderthequadraticfunctionanditsinverseontheentire

domain.Thisistoelicitadiscussionofwhenandunderwhatconditionstheinverseofafunctionis

alsoafunction.

CoreStandardsFocus:

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.

F.BF.4Findinversefunctions.

c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhas

aninverse.

d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.


SMP2–Reasonabstractlyandquantitatively

SMP7–Lookforandmakeuseofstructure

Vocabulary:Invertiblefunction




TheTeachingCycle:

Launch(WholeClass):

Beginthediscussionbysharingstudentideasaboutquestion#1.Somefactorsthattheymay

includeare:thekindoftiresonthecar,thetypeofroadsurface,theangleoftheroadsurface,the

weather,thespeedofthecar,etc.It’snotimportanttoidentifywhichfactorsactuallyare

considered;thequestionissimplytofosterthinkingabouttheproblemsituation.Explainto

studentsthatifyouaredrivingyouwanttobesurethatthereisplentyofroombetweenyouand

thecarinfrontofyoutobesurethatyoudon’thitthecarifyoubothhavetostopsuddenly.Ask

studentstoworkproblems1-4.

Explore(SmallGroup):

Monitorstudentsastheyareworkingtoseethattheyarenotspendingtoomuchtimeontheinitial

questions,especiallyquestiond.Thepurposeofquestiondistothinkabouttherateofchangeina

quadraticrelationshipversusalinearrelationship.Manypeoplemaintainlinearthinkingpatterns

evenwhentheyknowtherelationshipisnotlinear.Also,watchtoseethatstudentsareidentifying

andlabelinganappropriatescaleforthegraphofthefunctioninthiscontext.

Discuss(WholeClass):

Whenstudentshavecompletedtheirworkon#4,beginthediscussionwiththegraphofd(s).Ask

studentstodescribethemathematicalfeaturesofthegraphincluding:domain,range,minimum

value,xandy-intercepts,increasingordecreasing,etc.Askstudentstousethegraphtoprovide

evidencefortheiranswerstoquestion#3.Then,directstudentstocompletethetask.


Monitorstudentsastheyworktobesuretheyaresettinguptablesandapplyingappropriatescales

totheirgraphs.Listenforstatementsthatcanbesharedabouttheinverserelationshipsthatare

highlightedinthisprocess.Theyshouldbenoticingthattheindependentanddependentvariables

arebeingswitchedandhowthatisappearinginthetableandthegraph.Theyhavenotbeenasked

towriteanequationtomodelthegraph,althoughmanystudentsmayidentifythatitisasquare

rootfunction.Whenstudentsgettoquestions10and11,theymayneedalittlehelpininterpreting

thequestion.





Beginmuchliketheearlierdiscussionwiththetable,graphandfeaturesof𝑠(𝑑).Highlight

commentsfromthestudentsthathaverecognizedtheinverserelationshipbetweenthetwo

functionsandarticulatetheswitchbetweenthedependentandindependentvariables.Showthat

thismakestheinversefunctionareflectionoverthe𝑦 = 𝑥line(includingtheaxes).Besurethat

youaskstudentstocomparethedomainandrangeofeachofthefunctionsaspartofthe

justificationoftheinverserelationship.

Shiftthediscussiontoquestion#10andaskstudentstoaddtheadditionalpartofthegraphof

𝑑(𝑠)thatwouldbeincludedifthedomainisextendedtoallrealnumbers.Askstudentshowthat

wouldchangethegraphof𝑠(𝑑).Would𝑠(𝑑)beafunction?Studentsshouldbefamiliarwith

parabolasthathaveahorizontallineofsymmetry,andtobeabletoarticulatewhytheyarenot

functions.Askstudentshowtheymightbeabletousethegraphofafunctiontotellifitsinverse

willbeafunction.Thisshouldbringupsomeintuitivethinkingaboutone-to-onefunctionsandthe

ideathatifafunctionhasthesamey-valueformorethanonex-valuethentheinversewillnotbea

function.Showstudentshowthedomainwaslimitedinthecaseof𝑑(𝑠)tomakethefunction

invertible(sothattheinverseisafunction).Youmaywishtoshowafewfunctiongraphsandask

studentshowtheymightselectadomainsothatthefunctionisinvertible.




1.2



READY Topic:SolvingforavariableSolveforx.

1. 17 = 5𝑥 + 2

2.2𝑥( − 5 = 3𝑥( − 12𝑥 +31

3.11 = √2𝑥 + 1

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


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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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GO Topic:Usingfunctionnotationtoevaluateafunction

Thefunctions aredefinedbelow.

Calculatetheindicatedfunctionvalues.Simplifyyouranswers.

13.𝑓(7) 14.𝑓(−9) 15.𝑓(𝑠) 16.𝑓(𝑠 − 𝑡)

17.𝑔(7) 18.𝑔(−9) 19.𝑔(𝑠) 20.𝑔(𝑠 − 𝑡)

21.ℎ(7) 22.ℎ(−9) 23.ℎ(𝑠) 24.ℎ(𝑠 − 𝑡)

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

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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:

𝑑(𝑡) = 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?

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8. Considerthefunction𝑓(𝑥) = 2+ .A) Whatarethedomainandrangeof𝑓(𝑥)?Is𝑓(𝑥)invertible?

B) Graph𝑓(𝑥)and𝑓01(𝑥)onthegridbelow.

C) Whatarethedomainandrangeof𝑓01(𝑥)?

9. If𝑓(3) = 8,whatis𝑓01(8)?Howdoyouknow?

10. If𝑓 5167 = 1.414,whatis𝑓01(1.414)?Howdoyouknow?

11. If𝑓(𝑎) = 𝑏whatis𝑓01(𝑏)?Willyouranswerchangeiff(x)isadifferentfunction?Explain.

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1. 3 Tracking The Tortoise – Teacher Notes A Solidify Understanding Task

Purpose:

Thepurposeofthistaskistwo-fold:

1. Toextendtheideasofinversetoexponentialfunctions.

2. Todevelopinformalideasaboutlogarithmsbaseduponunderstandinginverses.(These

ideaswillbeextendedandformalizedinModuleII:LogarithmicFunctions)

ThefirstpartofthetaskbuildsonearlierworkinAlgebraIwithexponentialfunctions.Students

aregivenanexponentialcontext,thedistancetravelledforagiventime,andthenaskedtoreverse

theirthinkingtoconsiderthetimetravelledforagivendistance.Aftercreatingagraphofthe

situation,studentsareaskedtoconsidertheextendeddomainofallrealnumbersandthinkabout

howthataffectstheinversefunction.Questionsattheendofthetaskpressonunderstandingof

theinverserelationshipandtheideathatafunctionanditsinverse“undo”eachother,using

functionnotationandparticularvaluesofafunctionanditsinverse.

CoreStandardsFocus:

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F.BF.4Findinversefunctions.

c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhas

aninverse.

F.BF.5(+)Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethis

relationshiptosolveproblemsinvolvinglogarithmsandexponents.


SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP8–Lookforandexpressregularityinrepeatedreasoning




Vocabulary:logarithm(Thewordshouldonlybeintroducedinthistaskasthenameofthe

functionthatistheinverseofanexponentialfunction.Thedefinitionof“logarithm”willbe

formalizedandextendedinModule2.)

TheTeachingCycle:

Launch(WholeClass):

Beginthetaskbybeingsurethatstudentsunderstandtheproblemsituation.(Ifstudentshave

doneMVPAlgebraItheywillbefamiliarwiththecontext.)Askstudentstoworkindividuallyon

problems1and2.Brieflydiscusstheanswerstoquestion1andbesurethatstudentsknowhowto

usethemodelgiven.Then,askstudentstodescribewhattheyknowaboutd(t)beforetheygraph

thefunction.Besuretoincludefeaturessuchas:continuous,y-intercept(0,1),andalways

increasing.Usetechnologytodrawthegraphandconfirmtheirpredictionsaboutthegraph.


Havestudentsworkontherestofthetask.Questions3and4helptodrawthemintotheinverse

context,muchliketheworkintheprevioustwotasks.Ifstudentsareinitiallystuck,askthemwhat

strategiestheyusedtodrawthegraphinFlippingFerraris.Encouragetheuseoftablestogenerate

pointsonthegraph.Studentsshouldusetheirgraphtoestimateavalueforquestion#6—theydo

notyethaveexperiencewithlogarithmstosolveanequationofthistype.

Questions8-11becomemoreabstractandrelyontheirpreviousworkwithinverses.Youmaywish

tostartthediscussionwhenmoststudentshavefinishedquestion#7anddiscussproblems3-7,

beforedirectingstudentstoworkon8-11andthendiscussingthoseproblems.


Beginthediscussionbyaskingastudenttopresentatableofvaluesforthegraphoft(d).Ask

studentstodescribehowtheyselectedthevaluesofdtoelicittheideathattheychosepowersof2

becausetheywereeasytothinkaboutthetime.Thisstrategywillbecomeincreasinglyimportantin

ModuleII,soitisimportanttomakeitexplicithere.Drawthegraphoftheinversefunctiont(d)

andaskstudentshowitcomparestothegraphofd(t).Theyshouldagainnoticethereflectionover

𝑦 = 𝑥.Askstudentstocomparethedomainandrangeofd(t)withthedomainandrangeoft(d).

Asstudentsdiscussthegraphsandconcludethatt(d)istheinversefunctionofd(t),askthemto




begintogeneralizethepatternstheyseeaboutinversestoallfunctions.Somequestionsyoucould

askare:

• Willafunctionanditsinversealwaysbereflectionsoverthe𝑦 = 𝑥line?Why?

• Willthedomainofthefunctionalwaysbetherangeoftheinversefunction?Whyorwhy

not?

Movethediscussiontoquestion#8.Itisimportantinthisearlyexposuretotheideasoflogarithms

thatstudentsconsiderthepartofalogarithmicgraphbetweenx=0andx=1.Asyouaskstudents

todraw𝑓01(𝑥),considersomeparticularvaluesinthisregion.Usethestrategyoffindingpowersof

2togeneratevaluesfor𝑥 = 16, 1>, 1?.Usetheiranswersfromthepreviouspartofthediscussionto

extendthegraphandjustifytheverticalasymptoteatx=0.Tellstudentsthatthisisanewkindof

functionthatwillbeexploredindepthlater,butitiscalledalogarithmicfunction.Thisfunctionis

written:𝑦 = log6 𝑥.(Noneedtospendtoomuchtimeonthenotation.Itwillbesolidifiedlater.)

Completethediscussionwithquestions9-10.Askstudentstoshowhowtheycanusethegraphsof

𝑓(𝑥)and𝑓01(𝑥)tojustifytheiranswers.Emphasizequestion11andaskstudentstomakea

generalargumentbasedonthethreetasksandtheirknowledgeofinversessofar.Endthe

discussionbyintroducingamoreformaldefinitionofinversefunctionsasfollows:

Inmathematics,aninversefunctionisafunctionthat“reverses”or“undoes”anotherfunction.To

describethisrelationshipinsymbols,wesay,“Thefunction𝑔istheinverseoffunction𝑓ifandonly

if𝑓(𝑎) = 𝑏and𝑔(𝑏) = 𝑎.Usingxandy,wewouldwrite𝑓(𝑥) = 𝑦and𝑔(𝑦) = 𝑥.




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 𝑏(𝑡) = 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.DemonstratehowJackusedthemodel𝑏(𝑡) = 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.Usethetabletofind𝑓(7)and𝑓'$(11,364).13.Usethetabletofind𝑓(9)and𝑓'$(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(𝑥)I

18.g(a)

19.g(b&) 20.g(a + b) 21.hFf(𝑥)I

22.h(a)

23.h(b&) 24.h(a + b) 25.hF𝑔(𝑥)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

𝑓(𝑥) = 𝑥 + 8 𝑓)*(𝑥) = 𝑥 − 8

𝑥 = 7 7 7 + 8 = 15

Inwords:Subtract8fromtheresult

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1.

2.

Inwords:

Input Output

𝑓(𝑥) = 3𝑥 𝑓)*(𝑥) =

𝑥 = 7 7 3 ∙ 7 = 21

Input Output

𝑓(𝑥) = 𝑥/ 𝑓)*(𝑥) =

𝑥 = 7 7 7/ = 49

Inwords:

21




3.

4.

5.

Inwords:

Input Output

𝑓(𝑥) = 23 𝑓)*(𝑥) =

𝑥 = 7 7 24 = 128

Input Output

𝑓(𝑥) = 2𝑥 − 5 𝑓)*(𝑥) =

𝑥 = 7 7 2 ∙ 7 − 5 = 9

Input Output

𝑓(𝑥) = 𝑥 + 53

𝑓)*(𝑥) =

𝑥 = 7 7 7 + 53

= 4

Inwords:

22




6.

7.

Input Output

𝑓(𝑥) = (𝑥 − 3)/ 𝑓)*(𝑥) =

𝑥 = 7 7 (7 − 3)/ = 16

Input Output

𝑓(𝑥) = 4 − √𝑥 𝑓)*(𝑥) =

𝑥 = 7 7 4 − √7

Inwords:

Inwords:

Inwords:

23




8.

9.Eachoftheseproblemsbeganwithx=7.Whatis thedifferencebetweenthe𝑥usedin𝑓(𝑥)andthe𝑥used in𝑓)*(𝑥)?

10.In#6,couldanyvalueof𝑥beusedin𝑓(𝑥)andstillgivethesameoutputfrom𝑓)*(𝑥)?Explain.Whatabout#7?

11.Basedonyourworkinthistaskandtheothertasksinthismodulewhatrelationshipsdoyouseebetweenfunctionsandtheirinverses?

Inwords:

Input Output

𝑓(𝑥) = 23 − 10 𝑓)*(𝑥) =

𝑥 = 7 7 24 − 10 = 118

24




1.4 Pulling A Rabbit Out Of The Hat – Teacher Notes A Solidify Understanding Task Purpose:Thepurposeofthistaskistosolidifystudents’understandingoftherelationshipbetween

functionsandtheirinversesandtoformalizewritinginversefunctions.Inthetask,studentsare

givenafunctionandaparticularvalueforinputvalue𝑥,andthenaskedtodescribeandwritethe

functionthatthatwillproduceanoutputthatistheoriginal𝑥value.Thetaskreliesonstudents’

intuitiveunderstandingofinverseoperationssuchassubtraction“undoing”additionorsquare

roots“undoing”squaring.Therearetwoexponentialproblemswherestudentscandescribe

“undoing”anexponentialfunctionandtheteachercansupportthewritingoftheinversefunction

usinglogarithmicnotation.

CoreStandardsFocus:

F.BF.4.Findinversefunctions.

a. Solveanequationoftheform𝑓(𝑥) = 𝑐forasimplefunctionfthathasaninverseandwrite

anexpressionfortheinverse.Forexample,𝑓(𝑥) = 2𝑥<or𝑓(𝑥) = (𝑥 + 1)/(𝑥– 1)for

𝑥 ≠ 1.

b. (+)Verifybycompositionthatonefunctionistheinverseofanother.


SMP6–AttendtoprecisionSMP7–Lookforandmakeuseofstructure

TheTeachingCycle:Launch(WholeClass):Beginclassbyhavingstudentstrythenumbertrickatthebeginningofthetask.Aftertheytryitwiththeirownnumber,helpthemtotrackthroughtheoperationstoshowwhyitworksasfollows:

• Pickanumber 𝑥

• Add6 𝑥 + 6

• Multiplytheresultby2 2(𝑥 + 6) = 2𝑥 + 12

• Subtract12 2𝑥 + 12 − 12 = 2𝑥

• Divideby2 /3/= 𝑥

• Theansweristhenumberyoustartedwith! 𝑥




Thisshouldhighlighttheideathatinverseoperations“undo”eachother.Afunctionmayinvolve

morethanoneoperation,soiftheinversefunctionisto“undo”thefunction,itmayhavemorethan

oneoperationandthoseoperationsmayneedtobeperformedinaparticularorder.Tellstudents

thatinthistask,theywillbefindinginversefunctions,whichwillbedescribedinwordsandthen

symbolically.Workthroughtheexamplewiththeclassandthenletthemtalkwiththeirpartners

orgroupabouttherestoftheproblems.


Monitorstudentsastheyworktoseethattheyaremakingsenseoftheinverseoperationsand

consideringtheorderthatisneededonthefunctionsthatrequiretwosteps.Encouragethemto

describetheoperationsinthecorrectorderbeforetheywritetheinversefunctionsymbolically.

Becausethenotationforlogarithmicfunctionshasbarelybeenintroducedintheprevioustask,

studentsmaynotknowhowtowritetheinversefunctionfor#2and#8.Tellthemthatis

acceptableaslongastheyhavedescribedtheoperationfortheinverseinwords.Acceptinformal

expressionslike,“undotheexponential”,butchallengestudentsthatmaysaythattheinverseofthe

exponentialissomekindofroot,likean“xthroot”.

Asyoulistentostudentstalkingabouttheproblems,findoneortwoproblemsthataregenerating

controversyormisconceptionstodiscusswiththeentireclass.


Beginthediscussionwithproblems#4and#6.Askstudentstodescribetheinversefunctionin

wordsandthenhelptheclasstowritetheinversefunction.Thensupportstudentsinusinglog

notationfor#3withthefollowingstatements:

24 = 128 log/ 128 = 7

𝑓(𝑥) = 23 𝑓)*(𝑥) = log/ 𝑥

DiscusssomeoftheproblemsthatgeneratedcontroversyorconfusionduringtheExplorephase.

Endthediscussionbychallengingstudentstowritetheexpressionfor#8.Beforeworkingonthe

notation,askstudentstodescribetheinverseoperationsanddecidehowtheorderhastogoto

properlyunwindthefunction.Theyshouldsaythatyouneedtoadd10andthenundothe




exponential.Givethemsometimetothinkabouthowtousenotationtowritethatandthenask

studentstoofferideas.Theyshouldhaveseenfrompreviousproblemsthatthe+10needstogo

intotheargumentofthefunctionbecauseitneedstohappenbeforeyouundotheexponential.So,

thenotationshouldbe:

𝑓(𝑥) = 23 − 10 𝑓)*(𝑥) = log/( 𝑥 + 10)

24 − 10 = 118 log/(118 + 10) = 7(because24 = 128)

Makesurethatthereistimelefttodiscussquestions9,10and11.Forquestion#9and10,the

mainpointtohighlightistheideathattheoutputofthefunctionbecomestheinputfortheinverse

andviceversa.Thisiswhythedomainandrangeofthetwofunctionsareswitched(assuming

suitablevaluesforeach).Pressstudentstomakeageneralargumentthatthiswouldbetruefor

anyfunctionanditsinverse.

Question#11isanopportunitytosolidifyalltheideasaboutinversethathavebeenexploredinthe

unitbeforethepracticetask.Someideasthatshouldemerge:

• Afunctionanditsinverseundoeachother.

• Thereareinverseoperationslikeaddition/subtraction,multiplication/division,

squaring/squarerooting.Functionsandtheirinversesusetheseoperationstogetherand

theyneedtobeintherightorder.

• Forafunctiontobeinvertible,theinversemustalsobeafunction.(Thatmeansthatthe

originalfunctionmustbeone-to-one.)

• Thedomainofafunctioncanberestrictedtomakeitinvertible.

• Afunctionanditsinverselooklikereflectionsoverthe𝑦 = 𝑥line.(Becarefulofthis

statementbecauseofthewaytheaxeschangeunitsforthistobetrue.)

• Thedomain(suitably-restricted)ofafunctionistherangeoftheinversefunctionandvice

versa.

Finalizethediscussionoffeaturesofinversefunctionsbyintroducingamoreformaldefinitionof

inversefunctionsasfollows:

Inmathematics,aninversefunctionisafunctionthat“reverses”or“undoes”another

function.Todescribethisrelationshipinsymbols,wesay,“Thefunction𝑔istheinverseof




function𝑓ifandonlyif𝑓(𝑎) = 𝑏and𝑔(𝑏) = 𝑎.Using𝑥and𝑦,wewouldwrite𝑓(𝑥) = 𝑦

and𝑔(𝑦) = 𝑥.

Ifyouchoose,youcanclosetheclasswithonemorenumberpuzzleforstudentstofigureouton

theirown:

• Pickanumber

• Add2

• Squaretheresult

• Subtract4timestheoriginalnumber

• Subtract4fromthatresult

• Takethesquarerootofthenumberthatisleft

• Theansweristhenumberyoustartedwith.




1.4



READY Topic:PropertiesofexponentsUsetheproductruleorthequotientruletosimplify.Leaveallanswersinexponentialformwithonlypositiveexponents.

1. 3" ∙ 3$

2.7& ∙ 7" 3.10)* ∙ 10+ 4. 5- ∙ 5)"

5. 𝑝&𝑝$

6. 2" ∙ 2)0 ∙ 2 7. 𝑏22𝑏)$ 8. +3

+4

9.-5

-

10.03

05 11.+

67

+65 12. 8

69

83

SET Topic:Inversefunction13. Giventhefunctions𝑓(𝑥) = √𝑥 − 1𝑎𝑛𝑑𝑔(𝑥) = 𝑥& + 7:

a.Calculate𝑓(16)𝑎𝑛𝑑𝑔(3).

b.Write𝑓(16)asanorderedpair.

c.Write𝑔(3)asanorderedpair.

d.Whatdoyourorderedpairsfor𝑓(16)and𝑔(3)imply?

e.Find𝑓(25).

f.Basedonyouranswerfor𝑓(25),predict𝑔(4).

g.Find𝑔(4). Didyouranswermatchyourprediction?

h.Are𝑓(𝑥)𝑎𝑛𝑑𝑔(𝑥)inversefunctions? Justifyyouranswer.


25



1.4



Matchthefunctioninthefirstcolumnwithitsinverseinthesecondcolumn.

𝑓(𝑥) 𝑓)2(𝑥)16.𝑓(𝑥) = 3𝑥 + 5

a.𝑓)2(𝑥) = 𝑙𝑜𝑔$𝑥

17.𝑓(𝑥) = 𝑥$ b.𝑓)2(𝑥) = √𝑥9

18.𝑓(𝑥) = √𝑥 − 33 c.𝑓)2(𝑥) = M)$0

19.𝑓(𝑥) = 𝑥0 d.𝑓)2(𝑥) = M0− 5

20.𝑓(𝑥) = 5M e.𝑓)2(𝑥) = 𝑙𝑜𝑔0𝑥

21.𝑓(𝑥) = 3(𝑥 + 5) f.𝑓)2(𝑥) = 𝑥$ + 3

22.𝑓(𝑥) = 3M g.𝑓)2(𝑥) = √𝑥3

GO Topic:Compositefunctionsandinverses

Calculate𝒇O𝒈(𝒙)R𝒂𝒏𝒅𝒈O𝒇(𝒙)Rforeachpairoffunctions.

(Note:thenotation(𝑓 ∘ 𝑔)(𝑥)𝑎𝑛𝑑(𝑔 ∘ 𝑓)(𝑥)meansthesamethingas𝑓O𝑔(𝑥)R𝑎𝑛𝑑𝑔O𝑓(𝑥)R,

respectively.)

23.𝑓(𝑥) = 2𝑥 + 5𝑔(𝑥) = M)$

&

24.𝑓(𝑥) = (𝑥 + 2)0𝑔(𝑥) = √𝑥9 − 2

25.𝑓(𝑥) = 0*𝑥 + 6𝑔(𝑥) = *(M)")

0

26.𝑓(𝑥) = )0M+ 2𝑔(𝑥) = )0

M)&

26



1.4



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

a. b.

c. d.

27.Graphtheliney=xoneachofthegraphsabove.Whatdoyounotice?

28.Doyouthinkyourobservationsaboutthegraphsin#27hasanythingtodowiththe

answersyougotwhenyoufound𝑓O𝑔(𝑥)R𝑎𝑛𝑑𝑔O𝑓(𝑥)R?Explain.

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

interestingaboutthesetwotriangles?

30.Shadethe2trianglesingraphd.Aretheyinterestinginthesameway?Explain.

27




1.5 Inverse Universe A Practice Understanding Task Youandyourpartnerhaveeachbeengivenadifferentsetofcards.Theinstructionsare:

1. Selectacardandshowittoyourpartner.2. Worktogethertofindacardinyourpartner’sset

ofcardsthatrepresentstheinverseofthefunctionrepresentedonyourcard.3. Recordthecardsyouselectedandthereasonthatyouknowthattheyareinversesinthe

spacebelow.4. Repeattheprocessuntilallofthecardsarepairedup.

*Forthistaskonly,assumethatalltablesrepresentpointsonacontinuousfunction.

Pair1: _____________________ Justificationofinverserelationship:____________________________________





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29




1.5 Inverse Universe – Teacher Notes A Practice Understanding Task

TeacherNote:Itwillbehelpfultohavethecardsusedinthistaskcopiedandcutbefore

distributingthemtostudents.

Purpose:Thepurposeofthistaskisforstudentstobecomemorefluentinfindinginversesandto

increasetheirflexibilityinthinkingaboutinversefunctionsusingtables,graphs,equations,and

verbaldescriptions.

CoreStandardsFocus:

F.BF.4Findinversefunctions.

a. Solveanequationoftheform𝑓(𝑥) = 𝑐forasimplefunctionfthathasaninverseandwrite

anexpressionfortheinverse.Forexample,𝑓(𝑥) = 2𝑥7 or𝑓(𝑥) = (𝑥 + 1)/(𝑥– 1)for𝑥 ≠

1.

b. (+)Verifybycompositionthatonefunctionistheinverseofanother.

c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasan

inverse.

d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.

RelatedStandards:F.BF.5


SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP6–Attendtoprecision

TheTeachingCycle:

Launch(WholeClass):

Besurethattheentireclassunderstandstheinstructions.Emphasizethatitisimportantforthe

discussionthattheyhavefullyjustifiedtheirchoicesontherecordingsheet.Asktheclassforideas

abouthowtheycanjustifythattwofunctionsareinversesandrecordtheirresponses.Tell




studentstoreferencethelistandtobeveryspecificaboutthefunctionsthattheyareworkingwith

astheywritetheirjustifications.Fulljustificationshouldincludethetypeoffunctionsthatare

shownintherepresentationsandhowtheycanshowthateitherthe(𝑥, 𝑦)valuesareswitchedover

theentiredomainofeachofthefunctionsorthatthetwofunctionsundoeachother(by

composition—eitherformallyorinformallybydescribingtheoperationsinthenecessaryorder).


Monitorstudentsastheywork,listeningforparticularlychallenging,controversial,orinsightful

commentsgeneratedbythework.Besurethatstudentsarerecordingtheirjustificationsandthey

aresufficientlyspecifictothecardstheyhaveselected.Whilelisteningtostudents,identifyone

pairofstudentstopresentforeachfunctionanditsinverse.


Iftimepermits,youmaychoosetohave10differentpairsofstudentseachdemonstrateapairof

functionsandhowtheyjustifiedthattheywereinverses.Lookforvarietyintheirjustifications,

includingcheckingtoseeifthegraphsarereflections,verifyingbycompositionofspecificvalues,or

showinghowtheinputsandoutputshavebeenswitched.Ifthereisnotenoughtimetogothrough

eachpair,thenselectthecardsthatcausedthemostdiscussionduringtheexplorationphase.The

functionpairsare:

A1 B7A2 B3A3 B6A4 B8A5 B2A6 B1A7 B4A8 B5A9 B10A10 B9





A1

𝑓(𝑥) = &−2𝑥 − 2, −5 < 𝑥 < 0−2, 𝑥 ≥ 0

A3 Eachinputvalue,𝑥,issquaredandthen3isaddedtotheresult.Thedomainofthefunctionis[0,∞)

A5

x y-2 -32 30 06 54 4−43

-2

A2Thefunctionincreasesataconstantrateof0

1andthey-interceptis(0,c).

A4

A6

𝑦 = 33




A8

A7

x y-5 -125-3 -27-1 -11 13 275 125

A9

A10Yasminstartedasavingsaccountwith$5.Attheendofeachweek,sheadded3.Thisfunctionmodelstheamountofmoneyintheaccountforagivenweek.




B1

𝑦 = log7𝑥

B2

𝑓(𝑥) = 923𝑥, −3 < 𝑥 < 3

2𝑥 − 4, 𝑥 ≥ 3

B4

x y-216 -6-64 -4-8 -20 08 264 4216 6

B3Thex-interceptis(c,0)andtheslopeofthelineis1

0.

B6

x y3 04 17 212 319 428 539 6

B5




B7

B8

𝒙 y-2 -3-1 -20 11 62 13

B9

B10Thefunctioniscontinuousandgrowsbyanequalfactorof5overequalintervals.They-interceptis(0,1).



1.5



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

2.√𝑥' ∙ √𝑥( ∙ √𝑥)

3.√𝑎) ∙ √𝑎#' ∙ √𝑏&,

4.√32, ∙ √9 ∙ √27'

5.√8( ∙ √16' ∙ √2)

6.(5#)&

7.(7#)78

8.(379)7:

9.;:<(

:=>&

SET Topic:RepresentationsofinversefunctionsWritetheinverseofthegivenfunctioninthesameformatasthegivenfunction.Functionf(x)

Inverse𝑓78(𝑥)

10.x f(x)

-8 0

-4 3

0 6

4 9

8 12

10.


30



1.5



11. 12.

12.𝑓(𝑥) = −2𝑥 + 4

13.𝑓(𝑥) = 𝑙𝑜𝑔&𝑥

14.

15.x 𝑓(𝑥)

0 0

1 1

2 4

3 9

4 16

15.

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1.5



GO Topic:CompositefunctionsCalculate𝒇I𝒈(𝒙)L𝒂𝒏𝒅𝒈I𝒇(𝒙)Lforeachpairoffunctions.

(Note:thenotation(𝑓 ∘ 𝑔)(𝑥)𝑎𝑛𝑑(𝑔 ∘ 𝑓)(𝑥)meanthesamething,respectively.)

16.𝑓(𝑥) = 3𝑥 + 7; 𝑔(𝑥) = −4𝑥 − 11

17.𝑓(𝑥) = −4𝑥 + 60; 𝑔(𝑥) = − 89𝑥 + 15

18.𝑓(𝑥) = 10𝑥 − 5; 𝑔(𝑥) = #:𝑥 + 3

19.𝑓(𝑥) = −#&𝑥 + 4; 𝑔(𝑥) = − &

#𝑥 + 6

20.Lookbackatyourcalculationsfor𝑓I𝑔(𝑥)L𝑎𝑛𝑑𝑔I𝑓(𝑥)L.Twoofthepairsofequationsare

inversesofeachother.Whichonesdoyouthinktheyare?

Why?

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Functions and Their Inverses - Mathematics Vision Project · 2020-02-06 · ALGEBRA II // MODULE 1...

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Transcript of Functions and Their Inverses - Mathematics Vision Project · 2020-02-06 · ALGEBRA II // MODULE 1...