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The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2018 Mathematics Vision Project All Rights Reserved
MODULE 1
Functions & Their Inverses
SECONDARY
MATH THREE
An Integrated Approach
Standard Teacher Notes
SECONDARY MATH 3 // MODULE 1
FUNCTIONS AND THEIR INVERSES
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
MODULE 1 - TABLE OF CONTENTS
FUNCTIONS AND THEIR INVERSES
1.1Brutus Bites Back – A Develop Understanding Task Develops the concept of inverse functions in a linear modeling context using tables, graphs, and equations. (F.BF.1, F.BF.4, F.BF.4a) Ready, Set, Go Homework: Functions and Their Inverses 1.1
1.2Flipping Ferraris – A Solidify Understanding Task Extends the concepts of inverse functions in a quadratic modeling context with a focus on domain and range and whether a function is invertible in a given domain. (F.BF.1, F.BF.4, F.BF.4c, F.BF.4d) Ready, Set, Go Homework: Functions and Their Inverses 1.2
1.3Tracking the Tortoise – A Solidify Understanding Task Solidifies the concepts of inverse function in an exponential modeling context and surfaces ideas about logarithms. (F.BF.1, F.BF.4, F.BF.4c, F.BF.4d)Ready, Set, Go Homework: Functions and Their Inverses 1.3
1.4 Pulling a Rabbit Out of a Hat – A Solidify Understanding Task Uses function machines to model functions and their inverses. Focus on finding inverse functions and verifying that two functions are inverses. (F.BF.4, F.BF.4a, F.BF.4b) Ready, Set, Go Homework: Functions and Their Inverses 1.4
1.5 Inverse Universe – A Practice Understanding Task Uses tables, graphs, equations, and written descriptions of functions to match functions and their inverses together and to verify the inverse relationship between two functions. (F.BF.4a, F.BF.4b, F.BF.4c, F.BF.4d) Ready, Set, Go Homework: Functions and Their Inverses 1.5
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Claritathoughtabouthowmuchfoodtheycouldbuyforagivenamountofmoneyanddrewthisgraph:
2.WritetheequationofthefunctionthatClaritagraphed.
3. WriteaquestionthatwouldbemosteasilyansweredbyCarlos’graph.WriteaquestionthatwouldbemosteasilyansweredbyClarita’sgraph.Whatisthedifferencebetweenthetwoquestions?
4. Whatistherelationshipbetweenthetwofunctions?Howdoyouknow?
5.Usefunctionnotationtowritetherelationshipbetweenthefunctions.
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Lookingonline,Carlosfoundacompanythatwillsell8poundsofBrutusBitesfor$6plusaflat$5shippingchargeforeachorder.Thecompanyadvertisesthattheywillsellanyamountoffoodatthesamepriceperpound.
6. ModeltherelationshipbetweenthepriceandtheamountoffoodusingCarlos’approach.
7. ModeltherelationshipbetweenthepriceandtheamountoffoodusingClarita’sapproach.
8. Whatistherelationshipbetweenthesetwofunctions?Howdoyouknow?
9. Usefunctionnotationtowritetherelationshipbetweenthefunctions.
10. WhichcompanyshouldClaritaandCarlosbuytheirBrutusBitesfrom?Why?
3
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.1 Brutus Bites Back – Teacher Notes A Develop Understanding Task
Purpose:Thepurposeofthistaskistodevelopandextendtheconceptofinversefunctionsina
linearcontext.Inthetask,studentsusetables,graphs,andequationstorepresentinversefunctions
astwodifferentwaysofmodelingthesamesituation.Therepresentationsexposetheideathatthe
domainofthefunctionistherangeoftheinverse(andviceversa)forsuitablyrestricteddomains.
Studentsmayalsonoticethatthegraphsoftheinversefunctionsarereflectionsoverthe! = $line,butwiththeunderstandingthattheaxesarepartofthereflection.
CoreStandardsFocus:
F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F.BF.4Findinversefunctions.
a. Solveanequationoftheform%($) = (forasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.
RelatedStandards:
F-BF.4.b,F.BF.4c
StandardsforMathematicalPractice:
SMP1–Makesenseofproblemsandpersevereinsolvingthem
SMP7–Lookforandmakeuseofstructure
Vocabulary:Inversefunction
TheTeachingCycle:
Launch(WholeClass):Beginthetaskbyensuringthatstudentsunderstandthecontext.Ifthey
havenotdonethe“PetSitters”unitfromSecondaryMathematicsI,theymaynotbefamiliarwith
thestoryofCarlosandClarita,twostudentsthatstartedabusinesscaringforpets.Theimportant
partofthecontextissimplythattheyareplanningtobuydogfoodthatispricedbythepound.Ask
studentstobeginbyworkingindividuallyonquestions1,2,and3.Circulatearoundtheroom
lookingathowstudentsareworkingwiththegraphs.Identifystudentstopresenttotheclassthat
haveconstructedatableofvaluestoaidinwritingtheequation.Alsowatchtoseethevariables
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
thatstudentsareusingastheywriteequations.Iftheyareusing$astheindependentvariableforbothequations,thentheyhavenotproperlydefinedthevariable.Ifthisisacommonerror,you
maychoosetodiscussitwiththeclass.
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:)(*) = +,-./0+(0) =),-.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.
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Note:Problem#10isanextensionproblemforstudentsthatmaybeaheadoftheclassin
completingthetask.Itisintendedtogivestudentsanopportunityforadditionalanalysis,
butisnotcriticalforthedevelopmentoftheideasofinverseinthismodule.
AlignedReady,Set,Go:FunctionsandTheirInverses1.1
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
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1.1
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READY Topic:InverseoperationsInverseoperations“undo”eachother.Forinstance,additionandsubtractionareinverseoperations.
Soaremultiplicationanddivision.Inmathematics,itisoftenconvenienttoundoseveraloperations
inordertosolveforavariable.
Solveforxinthefollowingproblems.Thencompletethestatementbyidentifyingthe
operationyouusedto“undo”theequation.
1.24=3x Undomultiplicationby3by_______________________________________________
2. Undodivisionby5by______________________________________________________
3. Undoadd17by_____________________________________________________________
4. Undothesquarerootby___________________________________________________
5. Undothecuberootby_____________________________then___________________
6. Undoraisingxtothe4thpowerby________________________________________
7. Undosquaringby_______________________________then______________________
SET Topic:Linearfunctionsandtheirinverses
CarlosandClaritahaveapetsittingbusiness.Whentheyweretryingtodecidehowmanyeachof
dogsandcatstheycouldfitintotheiryard,theymadeatablebasedonthefollowinginformation.
Catpensrequire6ft2ofspace,whiledogrunsrequire24ft2.CarlosandClaritahaveupto360ft2
availableinthestorageshedforpensandruns,whilestillleavingenoughroomtomovearoundthe
cages.Theymadeatableofallofthecombinationsofcatsanddogstheycouldusetofillthespace.
Theyquicklyrealizedthattheycouldfitin4catsinthesamespaceasonedog.
cats 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60
dogs 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
x5= −2
x +17 = 20
x = 6
x +1( )3 = 2
x4 = 81
x − 9( )2 = 49
READY, SET, GO! Name PeriodDate
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SECONDARY MATH III // MODULE 1
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1.1
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8.Usetheinformationinthetabletowrite5orderedpairsthathavecatsastheinputvalue
anddogsastheoutputvalue.
9.Writeanexplicitequationthatshowshowmanydogstheycanaccommodatebasedonhow
manycatstheyhave.(Thenumberofdogs“d”willbeafunctionofthenumberofcats“c”or
! = #(%).)
10.Usetheinformationinthetabletowrite5orderedpairsthathavedogsastheinputvalue
andcatsastheoutputvalue.
11.Writeanexplicitequationthatshowshowmanycatstheycanaccommodatebasedonhow
manydogstheyhave.(Thenumberofcats“c”willbeafunctionofthenumberofdogs“d”
or% = '(!).)
Baseyouranswersin#12and#13onthetableatthetopofthepage.
12.Lookbackatproblem8andproblem10.Describehowtheorderedpairsaredifferent.
13.a)Lookbackattheequationyouwroteinproblem9.Describethedomainfor! = #(%).
b)Describethedomainfortheequation% = '(!)thatyouwroteinproblem11.
c)Whatistherelationshipbetweenthem?
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1.1
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GO Topic:Usingfunctionnotationtoevaluateafunction.
Thefunctions aredefinedbelow.
Calculatetheindicatedfunctionvaluesinthefollowingproblems.Simplifyyouranswers.
14.
15. 16. 17.
18.
19. 20. 20.'() + +)
22.
23. 24. 25.
f x( ), g x( ), and h x( )f x( ) = x g x( ) = 5x −12 h x( ) = x2 + 4x − 7
f 10( ) f −2( ) f a( ) f a + b( )
g 10( ) g −2( ) g a( )
h 10( ) h −2( ) h a( ) h a + b( )
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FUNCTIONS AND THEIR INVERSES – 1.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.2 Flipping Ferraris
A Solidify Understanding Task
Whenpeoplefirstlearntodrive,theyareoftentoldthatthefastertheyaredriving,thelongeritwilltaketostop.So,whenyou’redrivingonthefreeway,youshouldleavemorespacebetweenyourcarandthecarinfrontofyouthanwhenyouaredrivingslowlythroughaneighborhood.Haveyoueverwonderedabouttherelationshipbetweenhowfastyouaredrivingandhowfaryoutravelbeforeyoustop,afterhittingthebrakes?
1. Thinkaboutitforaminute.Whatfactorsdoyouthinkmightmakeadifferenceinhowfaracartravelsafterhittingthebrakes?
Therehasactuallybeenquiteabitofexperimentalworkdone(mostlybypolicedepartmentsandinsurancecompanies)tobeabletomathematicallymodeltherelationshipbetweenthespeedofacarandthebrakingdistance(howfarthecargoesuntilitstopsafterthedriverhitsthebrakes).
2. Imagineyourdreamcar.MaybeitisaFerrari550Maranello,asuper-fastItaliancar.Experimentshaveshownthatonsmooth,dryroads,therelationshipbetweenthebrakingdistance(d)andspeed(s)isgivenby!(#) = 0.03#) .Speedisgiveninmiles/hourandthedistanceisinfeet.a) Howmanyfeetshouldyouleavebetweenyouandthecarinfrontofyouifyouare
drivingtheFerrariat55mi/hr?
b) Whatdistanceshouldyoukeepbetweenyouandthecarinfrontofyouifyouaredrivingat100mi/hr?
c) Ifanaveragecarisabout16feetlong,abouthowmanycarlengthsshouldyouhavebetweenyouandthatcarinfrontofyouifyouaredriving100mi/hr?
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SECONDARY MATH III // MODULE 1
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Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
d) Itmakessensetoalotofpeoplethatifthecarismovingatsomespeedandthengoestwiceasfast,thebrakingdistancewillbetwiceasfar.Isthattrue?Explainwhyorwhynot.
3.Graphtherelationshipbetweenbrakingdistanced(s),andspeed(s),below.
4.AccordingtotheFerrariCompany,themaximumspeedofthecarisabout217mph.UsethistodescribeallthemathematicalfeaturesoftherelationshipbetweenbrakingdistanceandspeedfortheFerrarimodeledby!(#) = 0.03#) .
5.WhatifthedriveroftheFerrari550wascruisingalongandsuddenlyhitthebrakestostopbecauseshesawacatintheroad?Sheskiddedtoastop,andfortunately,missedthecat.Whenshegotoutofthecarshemeasuredtheskidmarksleftbythecarsothatsheknewthatherbrakingdistancewas31ft.
a)Howfastwasshegoingwhenshehitthebrakes?
b)Ifshedidn’tseethecatuntilshewas15feetaway,whatisthefastestspeedshecouldbetravelingbeforeshehitthebrakesifshewantstoavoidhittingthecat?
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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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FUNCTIONS AND THEIR INVERSES – 1.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
10.Considerthefunction!(#) = 0.03#)overthedomainofallrealnumbers,notjustthedomainofthisproblemsituation.Howdoesthegraphchangefromthegraphofd(s)inquestion#3?
11.Howdoeschangingthedomainofd(s)changethegraphoftheinverseofd(s)?
12.Istheinverseofd(s)afunction?Justifyyouranswer.
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SECONDARY MATH III // MODULE 1
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
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.2
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Discuss(WholeClass):
Beginmuchliketheearlierdiscussionwiththetable,graphandfeaturesof#(!).Highlightcommentsfromthestudentsthathaverecognizedtheinverserelationshipbetweenthetwo
functionsandarticulatetheswitchbetweenthedependentandindependentvariables.Showthat
thismakestheinversefunctionareflectionoverthe2 = 3line(includingtheaxes).Besurethatyouaskstudentstocomparethedomainandrangeofeachofthefunctionsaspartofthe
justificationoftheinverserelationship.
Shiftthediscussiontoquestion#10andaskstudentstoaddtheadditionalpartofthegraphof
!(#)thatwouldbeincludedifthedomainisextendedtoallrealnumbers.Askstudentshowthatwouldchangethegraphof#(!).Would#(!)beafunction?Studentsshouldbefamiliarwithparabolasthathaveahorizontallineofsymmetry,andtobeabletoarticulatewhytheyarenot
functions.Askstudentshowtheymightbeabletousethegraphofafunctiontotellifitsinverse
willbeafunction.Thisshouldbringupsomeintuitivethinkingaboutone-to-onefunctionsandthe
ideathatifafunctionhasthesamey-valueformorethanonex-valuethentheinversewillnotbea
function.Showstudentshowthedomainwaslimitedinthecaseof!(#)tomakethefunctioninvertible(sothattheinverseisafunction).Youmaywishtoshowafewfunctiongraphsandask
studentshowtheymightselectadomainsothatthefunctionisinvertible.
AlignedReady,Set,Go:FunctionsandTheirInverses1.2
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1.2
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READY Topic:SolvingforavariableSolveforx.
1. 17 = 5% + 2
2.2%( − 5 = 3%( − 12% + 313.11 = 2% + 1
4. %( + % − 2 = 25.−4 = 5% + 1, 6. 352, = 7%( + 9,
7.9/ = 243 8.5/ = 00(1 9.4/ = 0
2(
SET Topic:Exploringinversefunctions
10.Studentsweregivenasetofdatatograph.Aftertheyhadcompletedtheirgraphs,each
studentsharedhisgraphwithhisshoulderpartner.WhenEthanandEmmasaweach
other’sgraphs,theyexclaimedtogether,“Yourgraphiswrong!”Neithergraphiswrong.
ExplainwhatEthanandEmmahavedonewiththeirdata.
Ethan’sgraph
Emma’sgraph
READY, SET, GO! Name PeriodDate
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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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GO Topic:Usingfunctionnotationtoevaluateafunction
Thefunctions f x( ), g x( ), and h x( ) aredefinedbelow.
f x( ) = 3x g x( ) = 10x + 4 h x( ) = x2 − x
Calculatetheindicatedfunctionvalues.Simplifyyouranswers.
13.3 7 14.3 −9 15.3 4 16.3 4 − 5
17.6 7 18.6 −9 19.6 4 20.6 4 − 5
21.ℎ 7 22.ℎ −9 23.ℎ 4 24.ℎ 4 − 5
Noticethatthenotationf(g(x))isindicatingthatyoureplacexinf(x)withg(x).
Simplifythefollowing.
25.f(g(x)) 26.f(h(x)) 27.g(f(x))
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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:
!(#) = 2( (dinmetersandtinseconds)
Thetortoisefamilydecidestowatchtheracefromthesidelinessothattheycanseetheirdarlingtortoisesister,Shellie,provethevalueofpersistence.
1. Howfarawayfromthestartinglinemustthefamilybe,tobelocatedintherightplaceforShellietorunby5secondsafterthebeginningoftherace?After10seconds?
2. Describethegraphofd(t),Shellie’sdistanceattimet.Whataretheimportantfeaturesofd(t)?
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3. Ifthetortoisefamilyplanstowatchtheraceat64metersawayfromShellie’sstartingpoint,howlongwilltheyhavetowaittoseeShellierunpast?
4. HowlongmusttheywaittoseeShellierunbyiftheystand1024metersawayfromherstartingpoint?
5. DrawagraphthatshowshowlongthetortoisefamilywillwaittoseeShellierunbyatagivenlocationfromherstartingpoint.
6. HowlongmustthefamilywaittoseeShellierunbyiftheystand220metersawayfrom
herstartingpoint?
7. Whatistherelationshipbetweend(t)andthegraphthatyouhavejustdrawn?Howdidyouused(t)todrawthegraphin#5?
15
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
8. Considerthefunction)(*) = 2+ .A) Whatarethedomainandrangeof)(*)?Is)(*)invertible?
B) Graph)(*)and)01(*)onthegridbelow.
C) Whatarethedomainandrangeof)01(*)?
9. If)(3) = 8,whatis)01(8)?Howdoyouknow?
10. If) 5167 = 1.414,whatis)01(1.414)?Howdoyouknow?
11. If)(;) = <whatis)01(<)?Willyouranswerchangeiff(x)isadifferentfunction?Explain.
16
SECONDARY MATH III // MODULE 1
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)
ThefirstpartofthetaskbuildsonearlierworkinSecondaryIandIIwithexponentialfunctions.
Studentsaregivenanexponentialcontext,thedistancetravelledforagiventime,andthenaskedto
reversetheirthinkingtoconsiderthetimetravelledforagivendistance.Aftercreatingagraphof
thesituation,studentsareaskedtoconsidertheextendeddomainofallrealnumbersandthink
abouthowthataffectstheinversefunction.Questionsattheendofthetaskpresson
understandingoftheinverserelationshipandtheideathatafunctionanditsinverse“undo”each
other,usingfunctionnotationandparticularvaluesofafunctionanditsinverse.
CoreStandardsFocus:
F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F.BF.4Findinversefunctions.
c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhas
aninverse.
F.BF.5(+)Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethis
relationshiptosolveproblemsinvolvinglogarithmsandexponents.
StandardsforMathematicalPractice:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP8–Lookforandexpressregularityinrepeatedreasoning
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.3
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Vocabulary:logarithm(Thewordshouldonlybeintroducedinthistaskasthenameofthe
functionthatistheinverseofanexponentialfunction.Thedefinitionof“logarithm”willbe
formalizedandextendedinModule2.)
TheTeachingCycle:
Launch(WholeClass):
Beginthetaskbybeingsurethatstudentsunderstandtheproblemsituation.(Ifstudentshave
doneModuleIinSecondaryMathematicsII,theywillbefamiliarwiththecontext.)Askstudentsto
workindividuallyonproblems1and2.Brieflydiscusstheanswerstoquestion1andbesurethat
studentsknowhowtousethemodelgiven.Then,askstudentstodescribewhattheyknowabout
d(t)beforetheygraphthefunction.Besuretoincludefeaturessuchas:continuous,y-intercept
(0,1),andalwaysincreasing.Usetechnologytodrawthegraphandconfirmtheirpredictionsabout
thegraph.
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)
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
andaskstudentshowitcomparestothegraphofd(t).Theyshouldagainnoticethereflectionover
= = *.Askstudentstocomparethedomainandrangeofd(t)withthedomainandrangeoft(d).Asstudentsdiscussthegraphsandconcludethatt(d)istheinversefunctionofd(t),askthemto
begintogeneralizethepatternstheyseeaboutinversestoallfunctions.Somequestionsyoucould
askare:
• Willafunctionanditsinversealwaysbereflectionsoverthe= = *line?Why?• Willthedomainofthefunctionalwaysbetherangeoftheinversefunction?Whyorwhy
not?
Movethediscussiontoquestion#8.Itisimportantinthisearlyexposuretotheideasoflogarithms
thatstudentsconsiderthepartofalogarithmicgraphbetweenx=0andx=1.Asyouaskstudents
todraw)01(*),considersomeparticularvaluesinthisregion.Usethestrategyoffindingpowersof2togeneratevaluesfor* = 1
6 ,1> ,
1? .Usetheiranswersfromthepreviouspartofthediscussionto
extendthegraphandjustifytheverticalasymptoteatx=0.Tellstudentsthatthisisanewkindof
functionthatwillbeexploredindepthlater,butitiscalledalogarithmicfunction.Thisfunctionis
written:= = log6 *.(Noneedtospendtoomuchtimeonthenotation.Itwillbesolidifiedlater.)
Completethediscussionwithquestions9-10.Askstudentstoshowhowtheycanusethegraphsof
)(*)and)01(*)tojustifytheiranswers.Emphasizequestion11andaskstudentstomakeageneralargumentbasedonthethreetasksandtheirknowledgeofinversessofar.Endthe
discussionbyintroducingamoreformaldefinitionofinversefunctionsasfollows:
Inmathematics,aninversefunctionisafunctionthat“reverses”or“undoes”anotherfunction.To
describethisrelationshipinsymbols,wesay,“ThefunctionCistheinverseoffunction)ifandonlyif)(;) = <andC(<) = ;.Usingxandy,wewouldwrite)(*) = =andC(=) = *.
AlignedReady,Set,Go:FunctionsandTheirInverses1.3
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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 2(4) = 36. (b in feet and t in hours) Jackwasagoodmathstudent,soalthoughheneverlookedathisbeanstalkduringthenight,heusedthefunctiontocalculatehowtallitshouldbeasitgrew.Thetableontherightshowsthecalculationshemadeeveryhalfhour.Hence,Jackwasnotsurprisedwhen,inthemorning,hesawthatthetopofthebeanstalkhaddisappearedintotheclouds.
Time(hours) Height(feet)1 31.5 5.22 92.5 15.63 273.5 46.84 814.5 140.35 2435.5 420.96 7296.5 1,262.77 2,1877.5 3,7888 6,5618.5 11,3649 19,6839.5 34,09210 59,049
READY, SET, GO! Name PeriodDate
17
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FUNCTIONS AND INVERSES –
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1.3
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7.DemonstratehowJackusedthemodel2(4) = 36 tocalculatehowhighthebeanstalkwouldbeafter6hourshadpassed.(Youmayusethetablebutwritedownwhereyouwouldputthenumbersinthefunctionifyoudidn’thavethetable.)
8.Duringthatsamenight,aneighborwasplayingwithhisdrone.Itwasprogrammedtohoverat
243ft.Howmanyhourshadthebeanstalkbeengrowingwhenitwasashighasthedrone?9.Didyouusethetableinthesamewaytoanswer#8asyoudidtoanswer#7? Explain.10.WhileJackwasmakinghistable,hewaswonderinghowtallthebeanstalkwouldbeafterthe
magical10hourshadpassed.Hequicklytypedthefunctionintohiscalculatortofindout.WritetheequationJackwouldhavetypedintohiscalculator.
11.Commercialjetsflybetween30,000ft.and36,000ft.Abouthowmanyhoursofgrowingcould
passbeforethebeanstalkmightinterferewithcommercialaircraft?Explainhowyougotyouranswer.
12.Usethetabletofind9(7)and9'$(11,364).13.Usethetabletofind9(9)and9'$(9).13.Explainwhyit’spossibletoanswersomeofthequestionsabouttheheightofthebeanstalkby
justpluggingthenumbersintothefunctionruleandwhysometimesyoucanonlyusethetable.
18
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
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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(H)I
18.g(a)
19.g(b&) 20.g
(a + b) 21.hFf(H)I
22.h(a)
23.h(b&) 24.h
(a + b) 25.hFG(H)I
f x( ), g x( ), and h x( )
19
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.4 Pulling a Rabbit Out of the Hat A Solidify Understanding Task
Ihaveamagictrickforyou:
• Pickanumber,anynumber.• Add6• Multiplytheresultby2• Subtract12• Divideby2• Theansweristhenumberyoustartedwith!
Peopleareoftenmystifiedbysuchtricksbutthoseofuswhohavestudiedinverseoperationsandinversefunctionscaneasilyfigureouthowtheyworkandevencreateourownnumbertricks.Let’sgetstartedbyfiguringouthowinversefunctionsworktogether.
Foreachofthefollowingfunctionmachines,decidewhatfunctioncanbeusedtomaketheoutputthesameastheinputnumber.Describetheoperationinwordsandthenwriteitsymbolically.
Here’sanexample:
Input Output
!(#) = # + 8 !)*(#) = # − 8
# = 7 7 7 + 8 = 15
Inwords:Subtract8fromtheresult
CC B
Y Ch
ristia
n Ka
dlub
a
http
s://f
lic.k
r/p/
fwNc
q
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.
2.
3.
Inwords:
Input Output
!(#) = 2. !)*(#) =
# = 7 7 2/ = 128
Inwords:
Input Output
!(#) = 3# !)*(#) =
# = 7 7 3 ∙ 7 = 21
Input Output
!(#) = #3 !)*(#) =
# = 7 7 73 = 49
Inwords:
21
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
4.
5.
6.
Input Output
!(#) = 2# − 5 !)*(#) =
# = 7 7 2 ∙ 7 − 5 = 9
Input Output
!(#) = # + 53 !)*(#) =
# = 7 7 7 + 53 = 4
Input Output
!(#) = (# − 3)3 !)*(#) =
# = 7 7 (7 − 3)3 = 16
Inwords:
Inwords:
Inwords:
22
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7.
8.
9.Eachoftheseproblemsbeganwithx=7.Whatisthedifferencebetweenthe#usedin!(#)andthe#usedin!)*(#)?
10.In#6,couldanyvalueof#beusedin!(#)andstillgivethesameoutputfrom!)*(#)?Explain.Whatabout#7?
11.Basedonyourworkinthistaskandtheothertasksinthismodulewhatrelationshipsdoyouseebetweenfunctionsandtheirinverses?
Input Output
!(#) = 4 − √# !)*(#) =
# = 7 7 4 − √7
Inwords:
Inwords:
Input Output
!(#) = 2. − 10 !)*(#) =
# = 7 7 2/ − 10 = 118
23
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1. 4 Pulling A Rabbit Out Of The Hat – Teacher Notes
A Solidify Understanding Task
Purpose:Thepurposeofthistaskistosolidifystudents’understandingoftherelationshipbetween
functionsandtheirinversesandtoformalizewritinginversefunctions.Inthetask,studentsare
givenafunctionandaparticularvalueforinputvalue#,andthenaskedtodescribeandwritethefunctionthatthatwillproduceanoutputthatistheoriginal#value.Thetaskreliesonstudents’intuitiveunderstandingofinverseoperationssuchassubtraction“undoing”additionorsquare
roots“undoing”squaring.Therearetwoexponentialproblemswherestudentscandescribe
“undoing”anexponentialfunctionandtheteachercansupportthewritingoftheinversefunction
usinglogarithmicnotation.
CoreStandardsFocus:
F.BF.4.Findinversefunctions.
a. Solveanequationoftheform!(#) = ;forasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.Forexample,!(#) = 2#<or!(#) = (# + 1)/(#– 1)for# ≠ 1.
b. (+)Verifybycompositionthatonefunctionistheinverseofanother.
StandardsforMathematicalPractice:
SMP6–Attendtoprecision
SMP7–Lookforandmakeuseofstructure
TheTeachingCycle:
Launch(WholeClass):
Beginclassbyhavingstudentstrythenumbertrickatthebeginningofthetask.Aftertheytryit
withtheirownnumber,helpthemtotrackthroughtheoperationstoshowwhyitworksasfollows:
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
• Pickanumber #• Add6 # + 6• Multiplytheresultby2 2(# + 6) = 2# + 12• Subtract12 2# + 12 − 12 = 2#• Divideby2 3.
3 = #• Theansweristhenumberyoustartedwith! #
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:
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2/ = 128 log3 128 = 7
!(#) = 2. !)*(#) = log3 #
DiscusssomeoftheproblemsthatgeneratedcontroversyorconfusionduringtheExplorephase.
Endthediscussionbychallengingstudentstowritetheexpressionfor#8.Beforeworkingonthe
notation,askstudentstodescribetheinverseoperationsanddecidehowtheorderhastogoto
properlyunwindthefunction.Theyshouldsaythatyouneedtoadd10andthenundothe
exponential.Givethemsometimetothinkabouthowtousenotationtowritethatandthenask
studentstoofferideas.Theyshouldhaveseenfrompreviousproblemsthatthe+10needstogo
intotheargumentofthefunctionbecauseitneedstohappenbeforeyouundotheexponential.So,
thenotationshouldbe:
!(#) = 2. − 10 !)*(#) = log3( # + 10)
2/ − 10 = 118 log3(118 + 10) = 7(because2/ = 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.
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.4
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• AfunctionanditsinverselooklikereflectionsovertheD = #line.(Becarefulofthisstatementbecauseofthewaytheaxeschangeunitsforthistobetrue.)
• Thedomain(suitably-restricted)ofafunctionistherangeoftheinversefunctionandvice
versa.
Finalizethediscussionoffeaturesofinversefunctionsbyintroducingamoreformaldefinitionof
inversefunctionsasfollows:
Inmathematics,aninversefunctionisafunctionthat“reverses”or“undoes”another
function.Todescribethisrelationshipinsymbols,wesay,“ThefunctionEistheinverseoffunction!ifandonlyif!(F) = GandE(G) = F.Using#andD,wewouldwrite!(#) = DandE(D) = #.
Ifyouchoose,youcanclosetheclasswithonemorenumberpuzzleforstudentstofigureouton
theirown:
• Pickanumber
• Add2
• Squaretheresult
• Subtract4timestheoriginalnumber
• Subtract4fromthatresult
• Takethesquarerootofthenumberthatisleft
• Theansweristhenumberyoustartedwith.
AlignedReady,Set,Go:FunctionsandTheirInverses1.4
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FUNCTIONS AND INVERSES –
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READY Topic:PropertiesofexponentsUsetheproductruleorthequotientruletosimplify.Leaveallanswersinexponentialformwithonlypositiveexponents.
1. 3" ∙ 3$
2.7& ∙ 7" 3.10)* ∙ 10+ 4.5- ∙ 5)"
5..&.$
6.2" ∙ 2)0 ∙ 2 7.1221)$ 8.+3
+4
9.-5
-
10.03
05 11.
+67
+65 12.8
69
83
SET Topic:Inversefunction13. Giventhefunctions: ; = ; − 1?@AB ; = ;& + 7:
a.Calculate: 16 ?@AB 3 .
b.Write: 16 asanorderedpair.
c.WriteB 3 asanorderedpair.
d.Whatdoyourorderedpairsfor: 16 andB 3 imply?
e.Find: 25 .
f.Basedonyouranswerfor: 25 ,predictB 4 .
g.FindB 4 . Didyouranswermatchyourprediction?
h.Are: ; ?@AB ; inversefunctions? Justifyyouranswer.
READY, SET, GO! Name PeriodDate
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1.4
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Matchthefunctioninthefirstcolumnwithitsinverseinthesecondcolumn.
: ; :)2 ; 16.: ; = 3; + 5
a.:)2 ; = HIB$;
17.: ; = ;$ b.:)2 ; = ;9
18.: ; = ; − 33 c.:)2 ; =J)$
0
19.: ; = ;0 d.:)2 ; =J
0− 5
20.: ; = 5J e.:)2 ; = HIB0;
21.: ; = 3 ; + 5 f.:)2 ; = ;$ + 3
22.: ; = 3J g.:)2 ; = ;3
GO Topic:Compositefunctionsandinverses
CalculateK L M NOPL K M foreachpairoffunctions.
(Note:thenotation : ∘ B ; ?@A B ∘ : ; meansthesamethingas: B ; ?@AB : ; ,
respectively.)
23.: ; = 2; + 5B ; =J)$
&
24.: ; = ; + 2 0B ; = ;9 − 2
25.: ; =0
*; + 6B ; =
* J)"
0
26.: ; =)0
J+ 2B ; =
)0
J)&
25
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
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1.4
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Match the pairs of functions above (23-26) with their graphs. Label f (x) and g (x). a. b.
c. d.
27.Graphtheliney=xoneachofthegraphsabove.Whatdoyounotice?
28.Doyouthinkyourobservationsaboutthegraphsin#27hasanythingtodowiththe
answersyougotwhenyoufound: B ; ?@AB : ; ?Explain.
29.Lookatgraphb.Shadethe2trianglesmadebythey-axis,x-axis,andeachline.Whatis
interestingaboutthesetwotriangles?
30.Shadethe2trianglesingraphd.Aretheyinterestinginthesameway?Explain.
26
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org
1.5 Inverse Universe A Practice Understanding Task
Youandyourpartnerhaveeachbeengivenadifferent
setofcards.Theinstructionsare:
1. Selectacardandshowittoyourpartner.2. Worktogethertofindacardinyourpartner’sset
ofcardsthatrepresentstheinverseofthefunctionrepresentedonyourcard.
3. Recordthecardsyouselectedandthereasonthatyouknowthattheyareinversesinthespacebelow.
4. Repeattheprocessuntilallofthecardsarepairedup.
*Forthistaskonly,assumethatalltablesrepresentpointsonacontinuousfunction.
Pair1: _____________________ Justificationofinverserelationship:____________________________________
Pair2: _____________________ Justificationofinverserelationship:____________________________________
Pair3: _____________________ Justificationofinverserelationship:____________________________________
Pair4: _____________________ Justificationofinverserelationship:____________________________________
Pair5: _____________________ Justificationofinverserelationship:____________________________________
CC B
Y ag
uayo
_sam
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uUq2
eR
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org
Pair6: _____________________ Justificationofinverserelationship:____________________________________
Pair6: _____________________ Justificationofinverserelationship:____________________________________
Pair7: _____________________ Justificationofinverserelationship:____________________________________
Pair8: _____________________ Justificationofinverserelationship:____________________________________
Pair9: _____________________ Justificationofinverserelationship:____________________________________
Pair10:_____________________ Justificationofinverserelationship:____________________________________
28
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org
A1
!(#) = &−2# − 2, −5 < # < 0−2, # ≥ 0
A3
Eachinputvalue,#,issquaredandthen3isaddedtotheresult.Thedomainofthe
functionis[0,∞)
A5
x y-2 -3
2 3
0 0
6 5
4 4
−43
-2
A2
Thefunctionincreasesataconstant
rateof01andthey-interceptis(0,c).
A4
A6
2 = 33
29
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org
A8
A7
x y-5 -125
-3 -27
-1 -1
1 1
3 27
5 125
A9
A10
Yasminstartedasavingsaccountwith
$5.Attheendofeachweek,sheadded
3.Thisfunctionmodelstheamountof
moneyintheaccountforagivenweek.
30
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org
B1
2 = log7#
B2
!(#) = 923 #, −3 < # < 32# − 4, # ≥ 3
B4
x y-216 -6
-64 -4
-8 -2
0 0
8 2
64 4
216 6
B3
Thex-interceptis(c,0)andtheslope
ofthelineis10.
B6
x y3 0
4 1
7 2
12 3
19 4
28 5
39 6
B5
31
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org
B7
B8
: y-2 -3
-1 -2
0 1
1 6
2 13
B9
B10
Thefunctioniscontinuousandgrows
byanequalfactorof5overequal
intervals.They-interceptis(0,1).
32
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org
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. Solveanequationoftheform!(#) = ;forasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.Forexample,!(#) = 2#7 or!(#) = (# + 1)/(#– 1)for# ≠1.
b. (+)Verifybycompositionthatonefunctionistheinverseofanother.
c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasan
inverse.
d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.
RelatedStandards:F.BF.5
StandardsforMathematicalPractice:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP6–Attendtoprecision
Launch(WholeClass):
Besurethattheentireclassunderstandstheinstructions.Emphasizethatitisimportantforthe
discussionthattheyhavefullyjustifiedtheirchoicesontherecordingsheet.Asktheclassforideas
abouthowtheycanjustifythattwofunctionsareinversesandrecordtheirresponses.Tell
studentstoreferencethelistandtobeveryspecificaboutthefunctionsthattheyareworkingwith
astheywritetheirjustifications.Fulljustificationshouldincludethetypeoffunctionsthatare
shownintherepresentationsandhowtheycanshowthateitherthe(#, 2)valuesareswitchedover
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
© 2018 Mathematics Vision Project All Rights Reserved for the Additions and Enhancements mathematicsvisionproject.org
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 B7
A2 B3
A3 B6
A4 B8
A5 B2
A6 B1
A7 B4
A8 B5
A9 B10
A10 B9
AlignedReady,Set,Go:FunctionsandTheirInverses1.5
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.5
Needhelp?Visitwww.rsgsupport.org
READY Topic:PropertiesofexponentsUsepropertiesofexponentstosimplifythefollowing.Writeyouranswersinexponentialformwithpositiveexponents.1. !"
#∙ !%#
2. !& ∙ !' ∙ !(
3. )( ∙ )"
&∙ *%+
4. 32+ ∙ 9 ∙ 27&
5. 8' ∙ 16& ∙ 2(
6. 5" %
7. 7" 45
8. 346 47
9. 78'
79
%
SET Topic:RepresentationsofinversefunctionsWritetheinverseofthegivenfunctioninthesameformatasthegivenfunction.Functionf(x)
Inverse:45 !
10.x f(x)
-8 0
-4 3
0 6
4 9
8 12
10.
READY, SET, GO! Name PeriodDate
33
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.5
Needhelp?Visitwww.rsgsupport.org
11. 12.
12.:(!) = −2! + 4
13.: ! = BCD%!
14.
15.x : !
0 0
1 1
2 4
3 9
4 16
15.
34
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.5
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GO Topic:CompositefunctionsCalculateE F G HIJF E G foreachpairoffunctions.
(Note:thenotation : ∘ D ! )LM D ∘ : ! meanthesamething,respectively.)
16.: ! = 3! + 7; D ! = −4! − 11
17.: ! = −4! + 60; D ! = −5
6! + 15
18.: ! = 10! − 5; D ! ="
7! + 3
19.: ! = −"
%! + 4; D ! = −
%
"! + 6
20.Lookbackatyourcalculationsfor: D ! )LMD : ! .Twoofthepairsofequationsare
inversesofeachother.Whichonesdoyouthinktheyare?
Why?
35