functions and their inverses - mathematics vision project · 2020-02-06 · algebra ii // module 1...
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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
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
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
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Claritathoughtabouthowmuchfoodtheycouldbuyforagivenamountofmoneyanddrewthisgraph:
2.WritetheequationofthefunctionthatClaritagraphed.
3. WriteaquestionthatwouldbemosteasilyansweredbyCarlos’graph.WriteaquestionthatwouldbemosteasilyansweredbyClarita’sgraph.Whatisthedifferencebetweenthetwoquestions?
4. Whatistherelationshipbetweenthetwofunctions?Howdoyouknow?
5.Usefunctionnotationtowritetherelationshipbetweenthefunctions.
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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
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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
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
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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.
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
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Note:Problem#10isanextensionproblemforstudentsthatmaybeaheadoftheclassin
completingthetask.Itisintendedtogivestudentsanopportunityforadditionalanalysis,
butisnotcriticalforthedevelopmentoftheideasofinverseinthismodule.
AlignedReady,Set,Go:FunctionsandTheirInverses1.1
ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.1 1.1
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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
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.1
1.1
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.2
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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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FUNCTIONS AND THEIR INVERSES – 1.2
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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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FUNCTIONS AND THEIR INVERSES – 1.2
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.2
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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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FUNCTIONS AND THEIR INVERSES – 1.2
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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.
StandardsforMathematicalPractice:
SMP2–Reasonabstractlyandquantitatively
SMP7–Lookforandmakeuseofstructure
Vocabulary:Invertiblefunction
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.2
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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.
Explore(SmallGroup):
Monitorstudentsastheyworktobesuretheyaresettinguptablesandapplyingappropriatescales
totheirgraphs.Listenforstatementsthatcanbesharedabouttheinverserelationshipsthatare
highlightedinthisprocess.Theyshouldbenoticingthattheindependentanddependentvariables
arebeingswitchedandhowthatisappearinginthetableandthegraph.Theyhavenotbeenasked
towriteanequationtomodelthegraph,althoughmanystudentsmayidentifythatitisasquare
rootfunction.Whenstudentsgettoquestions10and11,theymayneedalittlehelpininterpreting
thequestion.
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.2
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Discuss(WholeClass):
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.
AlignedReady,Set,Go:FunctionsandTheirInverses1.2
ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.2
1.2
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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
READY, SET, GO! Name PeriodDate
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ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.2
1.2
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.2
1.2
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.3
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.3
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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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FUNCTIONS AND THEIR INVERSES – 1.3
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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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FUNCTIONS AND THEIR INVERSES – 1.3
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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.
StandardsforMathematicalPractice:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP8–Lookforandexpressregularityinrepeatedreasoning
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FUNCTIONS AND THEIR INVERSES – 1.3
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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.
Explore(SmallGroup):
Havestudentsworkontherestofthetask.Questions3and4helptodrawthemintotheinverse
context,muchliketheworkintheprevioustwotasks.Ifstudentsareinitiallystuck,askthemwhat
strategiestheyusedtodrawthegraphinFlippingFerraris.Encouragetheuseoftablestogenerate
pointsonthegraph.Studentsshouldusetheirgraphtoestimateavalueforquestion#6—theydo
notyethaveexperiencewithlogarithmstosolveanequationofthistype.
Questions8-11becomemoreabstractandrelyontheirpreviousworkwithinverses.Youmaywish
tostartthediscussionwhenmoststudentshavefinishedquestion#7anddiscussproblems3-7,
beforedirectingstudentstoworkon8-11andthendiscussingthoseproblems.
Discuss(WholeClass):
Beginthediscussionbyaskingastudenttopresentatableofvaluesforthegraphoft(d).Ask
studentstodescribehowtheyselectedthevaluesofdtoelicittheideathattheychosepowersof2
becausetheywereeasytothinkaboutthetime.Thisstrategywillbecomeincreasinglyimportantin
ModuleII,soitisimportanttomakeitexplicithere.Drawthegraphoftheinversefunctiont(d)
andaskstudentshowitcomparestothegraphofd(t).Theyshouldagainnoticethereflectionover
𝑦 = 𝑥.Askstudentstocomparethedomainandrangeofd(t)withthedomainandrangeoft(d).
Asstudentsdiscussthegraphsandconcludethatt(d)istheinversefunctionofd(t),askthemto
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.3
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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.
AlignedReady,Set,Go:FunctionsandTheirInverses1.3
ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.3
1.3
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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
READY, SET, GO! Name PeriodDate
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FUNCTIONS AND INVERSES – 1.3
1.3
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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.
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FUNCTIONS AND INVERSES – 1.3
1.3
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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( )
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FUNCTIONS AND THEIR INVERSES – 1.4
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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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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
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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
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FUNCTIONS AND THEIR INVERSES – 1.4
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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:
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ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
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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
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
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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
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FUNCTIONS AND THEIR INVERSES – 1.4
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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.
StandardsforMathematicalPractice:
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
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
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Thisshouldhighlighttheideathatinverseoperations“undo”eachother.Afunctionmayinvolve
morethanoneoperation,soiftheinversefunctionisto“undo”thefunction,itmayhavemorethan
oneoperationandthoseoperationsmayneedtobeperformedinaparticularorder.Tellstudents
thatinthistask,theywillbefindinginversefunctions,whichwillbedescribedinwordsandthen
symbolically.Workthroughtheexamplewiththeclassandthenletthemtalkwiththeirpartners
orgroupabouttherestoftheproblems.
Explore(SmallGroup):
Monitorstudentsastheyworktoseethattheyaremakingsenseoftheinverseoperationsand
consideringtheorderthatisneededonthefunctionsthatrequiretwosteps.Encouragethemto
describetheoperationsinthecorrectorderbeforetheywritetheinversefunctionsymbolically.
Becausethenotationforlogarithmicfunctionshasbarelybeenintroducedintheprevioustask,
studentsmaynotknowhowtowritetheinversefunctionfor#2and#8.Tellthemthatis
acceptableaslongastheyhavedescribedtheoperationfortheinverseinwords.Acceptinformal
expressionslike,“undotheexponential”,butchallengestudentsthatmaysaythattheinverseofthe
exponentialissomekindofroot,likean“xthroot”.
Asyoulistentostudentstalkingabouttheproblems,findoneortwoproblemsthataregenerating
controversyormisconceptionstodiscusswiththeentireclass.
Discuss(WholeClass):
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
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
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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
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
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function𝑓ifandonlyif𝑓(𝑎) = 𝑏and𝑔(𝑏) = 𝑎.Using𝑥and𝑦,wewouldwrite𝑓(𝑥) = 𝑦
and𝑔(𝑦) = 𝑥.
Ifyouchoose,youcanclosetheclasswithonemorenumberpuzzleforstudentstofigureouton
theirown:
• Pickanumber
• Add2
• Squaretheresult
• Subtract4timestheoriginalnumber
• Subtract4fromthatresult
• Takethesquarerootofthenumberthatisleft
• Theansweristhenumberyoustartedwith.
AlignedReady,Set,Go:FunctionsandTheirInverses1.4
ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.4
1.4
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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
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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.
READY, SET, GO! Name PeriodDate
25
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FUNCTIONS AND INVERSES – 1.4
1.4
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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
ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.4
1.4
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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
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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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:____________________________________
Pair2: _____________________ Justificationofinverserelationship:____________________________________
Pair3: _____________________ Justificationofinverserelationship:____________________________________
Pair4: _____________________ Justificationofinverserelationship:____________________________________
Pair5: _____________________ Justificationofinverserelationship:____________________________________
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28
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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Pair6: _____________________ Justificationofinverserelationship:____________________________________
Pair6: _____________________ Justificationofinverserelationship:____________________________________
Pair7: _____________________ Justificationofinverserelationship:____________________________________
Pair8: _____________________ Justificationofinverserelationship:____________________________________
Pair9: _____________________ Justificationofinverserelationship:____________________________________
Pair10:_____________________ Justificationofinverserelationship:____________________________________
29
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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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
StandardsforMathematicalPractice:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP6–Attendtoprecision
TheTeachingCycle:
Launch(WholeClass):
Besurethattheentireclassunderstandstheinstructions.Emphasizethatitisimportantforthe
discussionthattheyhavefullyjustifiedtheirchoicesontherecordingsheet.Asktheclassforideas
abouthowtheycanjustifythattwofunctionsareinversesandrecordtheirresponses.Tell
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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studentstoreferencethelistandtobeveryspecificaboutthefunctionsthattheyareworkingwith
astheywritetheirjustifications.Fulljustificationshouldincludethetypeoffunctionsthatare
shownintherepresentationsandhowtheycanshowthateitherthe(𝑥, 𝑦)valuesareswitchedover
theentiredomainofeachofthefunctionsorthatthetwofunctionsundoeachother(by
composition—eitherformallyorinformallybydescribingtheoperationsinthenecessaryorder).
Explore(SmallGroup):
Monitorstudentsastheywork,listeningforparticularlychallenging,controversial,orinsightful
commentsgeneratedbythework.Besurethatstudentsarerecordingtheirjustificationsandthey
aresufficientlyspecifictothecardstheyhaveselected.Whilelisteningtostudents,identifyone
pairofstudentstopresentforeachfunctionanditsinverse.
Discuss(WholeClass):
Iftimepermits,youmaychoosetohave10differentpairsofstudentseachdemonstrateapairof
functionsandhowtheyjustifiedthattheywereinverses.Lookforvarietyintheirjustifications,
includingcheckingtoseeifthegraphsarereflections,verifyingbycompositionofspecificvalues,or
showinghowtheinputsandoutputshavebeenswitched.Ifthereisnotenoughtimetogothrough
eachpair,thenselectthecardsthatcausedthemostdiscussionduringtheexplorationphase.The
functionpairsare:
A1 B7A2 B3A3 B6A4 B8A5 B2A6 B1A7 B4A8 B5A9 B10A10 B9
AlignedReady,Set,Go:FunctionsandTheirInverses1.5
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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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
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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A8
A7
x y-5 -125-3 -27-1 -11 13 275 125
A9
A10Yasminstartedasavingsaccountwith$5.Attheendofeachweek,sheadded3.Thisfunctionmodelstheamountofmoneyintheaccountforagivenweek.
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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B1
𝑦 = log7𝑥
B2
𝑓(𝑥) = 923𝑥, −3 < 𝑥 < 3
2𝑥 − 4, 𝑥 ≥ 3
B4
x y-216 -6-64 -4-8 -20 08 264 4216 6
B3Thex-interceptis(c,0)andtheslopeofthelineis1
0.
B6
x y3 04 17 212 319 428 539 6
B5
ALGEBRA II // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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B7
B8
𝒙 y-2 -3-1 -20 11 62 13
B9
B10Thefunctioniscontinuousandgrowsbyanequalfactorof5overequalintervals.They-interceptis(0,1).
ALGEBRA II // MODULE 1
FUNCTIONS AND INVERSES – 1.5
1.5
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READY Topic:PropertiesofexponentsUsepropertiesofexponentstosimplifythefollowing.Writeyouranswersinexponentialformwithpositiveexponents.1.√𝑥#$ ∙ √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.
READY, SET, GO! Name PeriodDate
30
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FUNCTIONS AND INVERSES – 1.5
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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.
31
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FUNCTIONS AND INVERSES – 1.5
1.5
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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?
32