Functions & Their Inverses - mathematicsvisionproject.org€¦ · The Mathematics Vision Project...

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


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.

CC B

Y Fr

anco

Van

nini

ht

tps:/

/flic

.kr/

p/m

EcoS

h

1




Claritathoughtabouthowmuchfoodtheycouldbuyforagivenamountofmoneyanddrewthisgraph:

2.WritetheequationofthefunctionthatClaritagraphed.

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

4. Whatistherelationshipbetweenthetwofunctions?Howdoyouknow?

5.Usefunctionnotationtowritetherelationshipbetweenthefunctions.

2




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

6. ModeltherelationshipbetweenthepriceandtheamountoffoodusingCarlos’approach.

7. ModeltherelationshipbetweenthepriceandtheamountoffoodusingClarita’sapproach.

8. Whatistherelationshipbetweenthesetwofunctions?Howdoyouknow?

9. Usefunctionnotationtowritetherelationshipbetweenthefunctions.

10. WhichcompanyshouldClaritaandCarlosbuytheirBrutusBitesfrom?Why?

3




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




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.




Note:Problem#10isanextensionproblemforstudentsthatmaybeaheadoftheclassin

completingthetask.Itisintendedtogivestudentsanopportunityforadditionalanalysis,

butisnotcriticalforthedevelopmentoftheideasofinverseinthismodule.

AlignedReady,Set,Go:FunctionsandTheirInverses1.1


FUNCTIONS AND INVERSES –


1.1

Needhelp?Visitwww.rsgsupport.org

READY Topic:InverseoperationsInverseoperations“undo”eachother.Forinstance,additionandsubtractionareinverseoperations.

Soaremultiplicationanddivision.Inmathematics,itisoftenconvenienttoundoseveraloperations

inordertosolveforavariable.

Solveforxinthefollowingproblems.Thencompletethestatementbyidentifyingthe

operationyouusedto“undo”theequation.

1.24=3x Undomultiplicationby3by_______________________________________________

2. Undodivisionby5by______________________________________________________

3. Undoadd17by_____________________________________________________________

4. Undothesquarerootby___________________________________________________

5. Undothecuberootby_____________________________then___________________

6. Undoraisingxtothe4thpowerby________________________________________

7. Undosquaringby_______________________________then______________________

SET Topic:Linearfunctionsandtheirinverses

CarlosandClaritahaveapetsittingbusiness.Whentheyweretryingtodecidehowmanyeachof

dogsandcatstheycouldfitintotheiryard,theymadeatablebasedonthefollowinginformation.

Catpensrequire6ft2ofspace,whiledogrunsrequire24ft2.CarlosandClaritahaveupto360ft2

availableinthestorageshedforpensandruns,whilestillleavingenoughroomtomovearoundthe

cages.Theymadeatableofallofthecombinationsofcatsanddogstheycouldusetofillthespace.

Theyquicklyrealizedthattheycouldfitin4catsinthesamespaceasonedog.

cats 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

dogs 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

x5= −2

x +17 = 20

x = 6

x +1( )3 = 2

x4 = 81

x − 9( )2 = 49

READY, SET, GO! Name PeriodDate

4




1.1


8.Usetheinformationinthetabletowrite5orderedpairsthathavecatsastheinputvalue

anddogsastheoutputvalue.

9.Writeanexplicitequationthatshowshowmanydogstheycanaccommodatebasedonhow

manycatstheyhave.(Thenumberofdogs“d”willbeafunctionofthenumberofcats“c”or

! = #(%).)

10.Usetheinformationinthetabletowrite5orderedpairsthathavedogsastheinputvalue

andcatsastheoutputvalue.

11.Writeanexplicitequationthatshowshowmanycatstheycanaccommodatebasedonhow

manydogstheyhave.(Thenumberofcats“c”willbeafunctionofthenumberofdogs“d”

or% = '(!).)

Baseyouranswersin#12and#13onthetableatthetopofthepage.

12.Lookbackatproblem8andproblem10.Describehowtheorderedpairsaredifferent.

13.a)Lookbackattheequationyouwroteinproblem9.Describethedomainfor! = #(%).

b)Describethedomainfortheequation% = '(!)thatyouwroteinproblem11.

c)Whatistherelationshipbetweenthem?

5




1.1


GO Topic:Usingfunctionnotationtoevaluateafunction.

Thefunctions aredefinedbelow.

Calculatetheindicatedfunctionvaluesinthefollowingproblems.Simplifyyouranswers.

14.

15. 16. 17.

18.

19. 20. 20.'() + +)

22.

23. 24. 25.

f x( ), g x( ), and h x( )f x( ) = x g x( ) = 5x −12 h x( ) = x2 + 4x − 7

f 10( ) f −2( ) f a( ) f a + b( )

g 10( ) g −2( ) g a( )

h 10( ) h −2( ) h a( ) h a + b( )

6




1.2 Flipping Ferraris

A Solidify Understanding Task

Whenpeoplefirstlearntodrive,theyareoftentoldthatthefastertheyaredriving,thelongeritwilltaketostop.So,whenyou’redrivingonthefreeway,youshouldleavemorespacebetweenyourcarandthecarinfrontofyouthanwhenyouaredrivingslowlythroughaneighborhood.Haveyoueverwonderedabouttherelationshipbetweenhowfastyouaredrivingandhowfaryoutravelbeforeyoustop,afterhittingthebrakes?

1. Thinkaboutitforaminute.Whatfactorsdoyouthinkmightmakeadifferenceinhowfaracartravelsafterhittingthebrakes?

Therehasactuallybeenquiteabitofexperimentalworkdone(mostlybypolicedepartmentsandinsurancecompanies)tobeabletomathematicallymodeltherelationshipbetweenthespeedofacarandthebrakingdistance(howfarthecargoesuntilitstopsafterthedriverhitsthebrakes).

2. Imagineyourdreamcar.MaybeitisaFerrari550Maranello,asuper-fastItaliancar.Experimentshaveshownthatonsmooth,dryroads,therelationshipbetweenthebrakingdistance(d)andspeed(s)isgivenby!(#) = 0.03#) .Speedisgiveninmiles/hourandthedistanceisinfeet.a) Howmanyfeetshouldyouleavebetweenyouandthecarinfrontofyouifyouare

drivingtheFerrariat55mi/hr?

b) Whatdistanceshouldyoukeepbetweenyouandthecarinfrontofyouifyouaredrivingat100mi/hr?

c) Ifanaveragecarisabout16feetlong,abouthowmanycarlengthsshouldyouhavebetweenyouandthatcarinfrontofyouifyouaredriving100mi/hr?

CC B

Y Da

rren

Price

ht

tps:/

/flic

.kr/

p/qv

LUQY

7




d) Itmakessensetoalotofpeoplethatifthecarismovingatsomespeedandthengoestwiceasfast,thebrakingdistancewillbetwiceasfar.Isthattrue?Explainwhyorwhynot.

3.Graphtherelationshipbetweenbrakingdistanced(s),andspeed(s),below.

4.AccordingtotheFerrariCompany,themaximumspeedofthecarisabout217mph.UsethistodescribeallthemathematicalfeaturesoftherelationshipbetweenbrakingdistanceandspeedfortheFerrarimodeledby!(#) = 0.03#) .

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

a)Howfastwasshegoingwhenshehitthebrakes?

b)Ifshedidn’tseethecatuntilshewas15feetaway,whatisthefastestspeedshecouldbetravelingbeforeshehitthebrakesifshewantstoavoidhittingthecat?

8




6.Partofthejobofpoliceofficersistoinvestigatetrafficaccidentstodeterminewhatcausedtheaccidentandwhichdriverwasatfault.Theymeasurethebrakingdistanceusingskidmarksandcalculatespeedsusingthemathematicalrelationshipsjustlikewehavehere,althoughtheyoftenusedifferentformulastoaccountforvariousfactorssuchasroadconditions.Let’sgobacktotheFerrarionasmooth,dryroadsinceweknowtherelationship.Createatablethatshowsthespeedthecarwastravelingbaseduponthebrakingdistance.

7.Writeanequationofthefunctions(d)thatgivesthespeedthecarwastravelingforagivenbrakingdistance.

8.Graphthefunctions(d)anddescribeitsfeatures.

9.Whatdoyounoticeaboutthegraphofs(d)comparedtothegraphofd(s)?Whatistherelationshipbetweenthefunctionsd(s)ands(d)?

9




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

11.Howdoeschangingthedomainofd(s)changethegraphoftheinverseofd(s)?

12.Istheinverseofd(s)afunction?Justifyyouranswer.

10




1.2 Flipping Ferraris – Teacher Notes A Solidify Understanding Task

SpecialNotetoTeachers:Studentsmaybeinterestedtoknowthattheinformationaboutthe

Ferrariprovidedinthistaskisbaseduponactualstudiesperformedtotestthecarsatvarious

speeds.Theunitsforspeedanddistancewerechangedfrommetrictothemorefamiliarimperial

unitssothatstudentscouldusetheirintuitiveknowledgetothinkaboutthereasonablenessoftheir

answers.

Purpose:Thepurposeofthistaskistoextendstudents’understandingofinversefunctionsto

includequadraticfunctionsandsquareroots.Inthetask,studentsaregiventheequationofa

quadraticfunctionforbrakingdistanceandaskedtothinkaboutthedistanceneededtostopsafely

foragivenspeed.Thelogicisthenturnedaroundandstudentsareaskedtomodelthespeedthe

carwasgoingforagivenbrakingdistance.Afterstudentshaveworkedwiththeinverse

relationshipswiththelimiteddomain[0,217](assumingthat217isthemaximumspeedofthe

car),thenstudentsareaskedtoconsiderthequadraticfunctionanditsinverseontheentire

domain.Thisistoelicitadiscussionofwhenandunderwhatconditionstheinverseofafunctionis

alsoafunction.

CoreStandardsFocus:

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.

F.BF.4Findinversefunctions.

c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhas

aninverse.

d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.


SMP2–Reasonabstractlyandquantitatively


Vocabulary:Invertiblefunction




TheTeachingCycle:

Launch(WholeClass):

Beginthediscussionbysharingstudentideasaboutquestion#1.Somefactorsthattheymay

includeare:thekindoftiresonthecar,thetypeofroadsurface,theangleoftheroadsurface,the

weather,thespeedofthecar,etc.It’snotimportanttoidentifywhichfactorsactuallyare

considered;thequestionissimplytofosterthinkingabouttheproblemsituation.Explainto

studentsthatifyouaredrivingyouwanttobesurethatthereisplentyofroombetweenyouand

thecarinfrontofyoutobesurethatyoudon’thitthecarifyoubothhavetostopsuddenly.Ask

studentstoworkproblems1-4.

Explore(SmallGroup):

Monitorstudentsastheyareworkingtoseethattheyarenotspendingtoomuchtimeontheinitial

questions,especiallyquestiond.Thepurposeofquestiondistothinkabouttherateofchangeina

quadraticrelationshipversusalinearrelationship.Manypeoplemaintainlinearthinkingpatterns

evenwhentheyknowtherelationshipisnotlinear.Also,watchtoseethatstudentsareidentifying

andlabelinganappropriatescaleforthegraphofthefunctioninthiscontext.

Discuss(WholeClass):

Whenstudentshavecompletedtheirworkon#4,beginthediscussionwiththegraphofd(s).Ask

studentstodescribethemathematicalfeaturesofthegraphincluding:domain,range,minimum

value,xandy-intercepts,increasingordecreasing,etc.Askstudentstousethegraphtoprovide

evidencefortheiranswerstoquestion#3.Then,directstudentstocompletethetask.


Monitorstudentsastheyworktobesuretheyaresettinguptablesandapplyingappropriatescales

totheirgraphs.Listenforstatementsthatcanbesharedabouttheinverserelationshipsthatare

highlightedinthisprocess.Theyshouldbenoticingthattheindependentanddependentvariables

arebeingswitchedandhowthatisappearinginthetableandthegraph.Theyhavenotbeenasked

towriteanequationtomodelthegraph,althoughmanystudentsmayidentifythatitisasquare

rootfunction.Whenstudentsgettoquestions10and11,theymayneedalittlehelpininterpreting

thequestion.





Beginmuchliketheearlierdiscussionwiththetable,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.





1.2


READY Topic:SolvingforavariableSolveforx.

1. 17 = 5% + 2

2.2%( − 5 = 3%( − 12% + 313.11 = 2% + 1

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

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

2(

SET Topic:Exploringinversefunctions

10.Studentsweregivenasetofdatatograph.Aftertheyhadcompletedtheirgraphs,each

studentsharedhisgraphwithhisshoulderpartner.WhenEthanandEmmasaweach

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

ExplainwhatEthanandEmmahavedonewiththeirdata.

Ethan’sgraph

Emma’sgraph


11




1.2


11.DescribeasequenceoftransformationsthatwouldtakeEthan’sgraphontoEmma’s.

12.Abaseballishitupwardfromaheightof3feetwith

aninitialvelocityof80feetpersecond(about55mph).Thegraphshowstheheightoftheballatanygivensecondduringitsflight.Usethegraphtoanswerthequestionsbelow.

a. Approximatethetimethattheballisatitsmaximumheight.

b. Approximatethetimethattheballhitstheground.

c. Atwhattimeistheball67feetabovetheground?

d. Makeanewgraphthatshowsthetimewhentheballisatthegivenheights.

e. Isyournewgraphafunction?Explain.

12




1.2


GO Topic:Usingfunctionnotationtoevaluateafunction

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

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

Calculatetheindicatedfunctionvalues.Simplifyyouranswers.

13.3 7 14.3 −9 15.3 4 16.3 4 − 5

17.6 7 18.6 −9 19.6 4 20.6 4 − 5

21.ℎ 7 22.ℎ −9 23.ℎ 4 24.ℎ 4 − 5

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

Simplifythefollowing.

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

13




1.3 Tracking the Tortoise


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)?

CC B

Y Sc

ot N

elso

n ht

tps:/

/flic

.kr/

p/M

5GzR

d

14




3. Ifthetortoisefamilyplanstowatchtheraceat64metersawayfromShellie’sstartingpoint,howlongwilltheyhavetowaittoseeShellierunpast?

4. HowlongmusttheywaittoseeShellierunbyiftheystand1024metersawayfromherstartingpoint?

5. DrawagraphthatshowshowlongthetortoisefamilywillwaittoseeShellierunbyatagivenlocationfromherstartingpoint.

6. HowlongmustthefamilywaittoseeShellierunbyiftheystand220metersawayfrom

herstartingpoint?

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

15




8. Considerthefunction)(*) = 2+ .A) Whatarethedomainandrangeof)(*)?Is)(*)invertible?

B) Graph)(*)and)01(*)onthegridbelow.

C) Whatarethedomainandrangeof)01(*)?

9. If)(3) = 8,whatis)01(8)?Howdoyouknow?

10. If) 5167 = 1.414,whatis)01(1.414)?Howdoyouknow?

11. If)(;) = <whatis)01(<)?Willyouranswerchangeiff(x)isadifferentfunction?Explain.

16




1. 3 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.


SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP8–Lookforandexpressregularityinrepeatedreasoning




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.


Havestudentsworkontherestofthetask.Questions3and4helptodrawthemintotheinverse

context,muchliketheworkintheprevioustwotasks.Ifstudentsareinitiallystuck,askthemwhat

strategiestheyusedtodrawthegraphinFlippingFerraris.Encouragetheuseoftablestogenerate

pointsonthegraph.Studentsshouldusetheirgraphtoestimateavalueforquestion#6—theydo

notyethaveexperiencewithlogarithmstosolveanequationofthistype.

Questions8-11becomemoreabstractandrelyontheirpreviousworkwithinverses.Youmaywish

tostartthediscussionwhenmoststudentshavefinishedquestion#7anddiscussproblems3-7,

beforedirectingstudentstoworkon8-11andthendiscussingthoseproblems.


Beginthediscussionbyaskingastudenttopresentatableofvaluesforthegraphoft(d).Ask

studentstodescribehowtheyselectedthevaluesofdtoelicittheideathattheychosepowersof2

becausetheywereeasytothinkaboutthetime.Thisstrategywillbecomeincreasinglyimportantin

ModuleII,soitisimportanttomakeitexplicithere.Drawthegraphoftheinversefunctiont(d)




andaskstudentshowitcomparestothegraphofd(t).Theyshouldagainnoticethereflectionover

= = *.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(=) = *.





1.3


READY Topic:Solvingexponentialequations.Solveforthevalueofx.

1. 5"#$ = 5&"'( 2. 7("'& = 7'&"#* 3. 4(" = 2&"'*

4. 3."'/ = 9&"'( 5. 8"#$ = 2&"#( 6. 3"#$ = $

*$

SET Topic:ExploringtheinverseofanexponentialfunctionInthefairytaleJackandtheBeanstalk,Jackplantsamagicbeanbeforehegoestobed.InthemorningJackdiscoversagiantbeanstalkthathasgrownsolarge,itdisappearsintotheclouds.Buthereisthepartofthestoryyouneverheard.Writtenonthebagcontainingthemagicbeanswasthisnote.Plant a magic bean in rich soil just as the sun is setting. Do not look at the plant site for 10 hours. (This is part of the magic.) After the bean has been in the ground for 1 hour, the growth of the sprout can be modeled by the function 2(4) = 36. (b in feet and t in hours) Jackwasagoodmathstudent,soalthoughheneverlookedathisbeanstalkduringthenight,heusedthefunctiontocalculatehowtallitshouldbeasitgrew.Thetableontherightshowsthecalculationshemadeeveryhalfhour.Hence,Jackwasnotsurprisedwhen,inthemorning,hesawthatthetopofthebeanstalkhaddisappearedintotheclouds.

Time(hours) Height(feet)1 31.5 5.22 92.5 15.63 273.5 46.84 814.5 140.35 2435.5 420.96 7296.5 1,262.77 2,1877.5 3,7888 6,5618.5 11,3649 19,6839.5 34,09210 59,049


17




1.3


7.DemonstratehowJackusedthemodel2(4) = 36 tocalculatehowhighthebeanstalkwouldbeafter6hourshadpassed.(Youmayusethetablebutwritedownwhereyouwouldputthenumbersinthefunctionifyoudidn’thavethetable.)

8.Duringthatsamenight,aneighborwasplayingwithhisdrone.Itwasprogrammedtohoverat

243ft.Howmanyhourshadthebeanstalkbeengrowingwhenitwasashighasthedrone?9.Didyouusethetableinthesamewaytoanswer#8asyoudidtoanswer#7? Explain.10.WhileJackwasmakinghistable,hewaswonderinghowtallthebeanstalkwouldbeafterthe

magical10hourshadpassed.Hequicklytypedthefunctionintohiscalculatortofindout.WritetheequationJackwouldhavetypedintohiscalculator.

11.Commercialjetsflybetween30,000ft.and36,000ft.Abouthowmanyhoursofgrowingcould

passbeforethebeanstalkmightinterferewithcommercialaircraft?Explainhowyougotyouranswer.

12.Usethetabletofind9(7)and9'$(11,364).13.Usethetabletofind9(9)and9'$(9).13.Explainwhyit’spossibletoanswersomeofthequestionsabouttheheightofthebeanstalkby

justpluggingthenumbersintothefunctionruleandwhysometimesyoucanonlyusethetable.

18




1.3


GO Topic:EvaluatingfunctionsThefunctions aredefinedbelow.

f(x) = −2x g(x) = 2x + 5 h(x) = x& + 3x − 10

Calculatetheindicatedfunctionvalues.Simplifyyouranswers.

14.f(a)

15.f(b&) 16.f(a + b) 17.fFG(H)I

18.g(a)

19.g(b&) 20.g

(a + b) 21.hFf(H)I

22.h(a)

23.h(b&) 24.h

(a + b) 25.hFG(H)I

f x( ), g x( ), and h x( )

19




1.4 Pulling a Rabbit Out of the Hat A Solidify Understanding Task

Ihaveamagictrickforyou:

• Pickanumber,anynumber.• Add6• Multiplytheresultby2• Subtract12• Divideby2• Theansweristhenumberyoustartedwith!

Peopleareoftenmystifiedbysuchtricksbutthoseofuswhohavestudiedinverseoperationsandinversefunctionscaneasilyfigureouthowtheyworkandevencreateourownnumbertricks.Let’sgetstartedbyfiguringouthowinversefunctionsworktogether.

Foreachofthefollowingfunctionmachines,decidewhatfunctioncanbeusedtomaketheoutputthesameastheinputnumber.Describetheoperationinwordsandthenwriteitsymbolically.

Here’sanexample:

Input Output

!(#) = # + 8 !)*(#) = # − 8

# = 7 7 7 + 8 = 15

Inwords:Subtract8fromtheresult

CC B

Y Ch

ristia

n Ka

dlub

a

http

s://f

lic.k

r/p/

fwNc

q

20




1.

2.

3.

Inwords:

Input Output

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

# = 7 7 2/ = 128

Inwords:

Input Output

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

# = 7 7 3 ∙ 7 = 21

Input Output

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

# = 7 7 73 = 49

Inwords:

21




4.

5.

6.

Input Output

!(#) = 2# − 5 !)*(#) =

# = 7 7 2 ∙ 7 − 5 = 9

Input Output

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

# = 7 7 7 + 53 = 4

Input Output

!(#) = (# − 3)3 !)*(#) =

# = 7 7 (7 − 3)3 = 16

Inwords:

Inwords:

Inwords:

22




7.

8.

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

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

11.Basedonyourworkinthistaskandtheothertasksinthismodulewhatrelationshipsdoyouseebetweenfunctionsandtheirinverses?

Input Output

!(#) = 4 − √# !)*(#) =

# = 7 7 4 − √7

Inwords:

Inwords:

Input Output

!(#) = 2. − 10 !)*(#) =

# = 7 7 2/ − 10 = 118

23




1. 4 Pulling A Rabbit Out Of The Hat – Teacher Notes


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.


SMP6–Attendtoprecision


TheTeachingCycle:

Launch(WholeClass):

Beginclassbyhavingstudentstrythenumbertrickatthebeginningofthetask.Aftertheytryit

withtheirownnumber,helpthemtotrackthroughtheoperationstoshowwhyitworksasfollows:




• 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.


Monitorstudentsastheyworktoseethattheyaremakingsenseoftheinverseoperationsand

consideringtheorderthatisneededonthefunctionsthatrequiretwosteps.Encouragethemto

describetheoperationsinthecorrectorderbeforetheywritetheinversefunctionsymbolically.

Becausethenotationforlogarithmicfunctionshasbarelybeenintroducedintheprevioustask,

studentsmaynotknowhowtowritetheinversefunctionfor#2and#8.Tellthemthatis

acceptableaslongastheyhavedescribedtheoperationfortheinverseinwords.Acceptinformal

expressionslike,“undotheexponential”,butchallengestudentsthatmaysaythattheinverseofthe

exponentialissomekindofroot,likean“xthroot”.

Asyoulistentostudentstalkingabouttheproblems,findoneortwoproblemsthataregenerating

controversyormisconceptionstodiscusswiththeentireclass.


Beginthediscussionwithproblems#4and#6.Askstudentstodescribetheinversefunctionin

wordsandthenhelptheclasstowritetheinversefunction.Thensupportstudentsinusinglog

notationfor#3withthefollowingstatements:




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.




• 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.





1.4


READY Topic:PropertiesofexponentsUsetheproductruleorthequotientruletosimplify.Leaveallanswersinexponentialformwithonlypositiveexponents.

1. 3" ∙ 3$

2.7& ∙ 7" 3.10)* ∙ 10+ 4.5- ∙ 5)"

5..&.$

6.2" ∙ 2)0 ∙ 2 7.1221)$ 8.+3

+4

9.-5

-

10.03

05 11.

+67

+65 12.8

69

83

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

a.Calculate: 16 ?@AB 3 .

b.Write: 16 asanorderedpair.

c.WriteB 3 asanorderedpair.

d.Whatdoyourorderedpairsfor: 16 andB 3 imply?

e.Find: 25 .

f.Basedonyouranswerfor: 25 ,predictB 4 .

g.FindB 4 . Didyouranswermatchyourprediction?

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


24




1.4


Matchthefunctioninthefirstcolumnwithitsinverseinthesecondcolumn.

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

a.:)2 ; = HIB$;

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

18.: ; = ; − 33 c.:)2 ; =J)$

0

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

0− 5

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

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

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

GO Topic:Compositefunctionsandinverses

CalculateK L M NOPL K M foreachpairoffunctions.

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

respectively.)

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

&

24.: ; = ; + 2 0B ; = ;9 − 2

25.: ; =0

*; + 6B ; =

* J)"

0

26.: ; =)0

J+ 2B ; =

)0

J)&

25




1.4


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

c. d.

27.Graphtheliney=xoneachofthegraphsabove.Whatdoyounotice?

28.Doyouthinkyourobservationsaboutthegraphsin#27hasanythingtodowiththe

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

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

interestingaboutthesetwotriangles?

30.Shadethe2trianglesingraphd.Aretheyinterestinginthesameway?Explain.

26



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

1.5 Inverse Universe A Practice Understanding Task

Youandyourpartnerhaveeachbeengivenadifferent

setofcards.Theinstructionsare:

1. Selectacardandshowittoyourpartner.2. Worktogethertofindacardinyourpartner’sset

ofcardsthatrepresentstheinverseofthefunctionrepresentedonyourcard.

3. Recordthecardsyouselectedandthereasonthatyouknowthattheyareinversesinthespacebelow.

4. Repeattheprocessuntilallofthecardsarepairedup.

*Forthistaskonly,assumethatalltablesrepresentpointsonacontinuousfunction.

Pair1: _____________________ Justificationofinverserelationship:____________________________________





CC B

Y ag

uayo

_sam

uel

http

s://f

lic.k

r/p/

uUq2

eR

27









Pair10:_____________________ Justificationofinverserelationship:____________________________________

28




A1

!(#) = &−2# − 2, −5 < # < 0−2, # ≥ 0

A3

Eachinputvalue,#,issquaredandthen3isaddedtotheresult.Thedomainofthe

functionis[0,∞)

A5

x y-2 -3

2 3

0 0

6 5

4 4

−43

-2

A2

Thefunctionincreasesataconstant

rateof01andthey-interceptis(0,c).

A4

A6

2 = 33

29




A8

A7

x y-5 -125

-3 -27

-1 -1

1 1

3 27

5 125

A9

A10

Yasminstartedasavingsaccountwith

$5.Attheendofeachweek,sheadded

3.Thisfunctionmodelstheamountof

moneyintheaccountforagivenweek.

30




B1

2 = log7#

B2

!(#) = 923 #, −3 < # < 32# − 4, # ≥ 3

B4

x y-216 -6

-64 -4

-8 -2

0 0

8 2

64 4

216 6

B3

Thex-interceptis(c,0)andtheslope

ofthelineis10.

B6

x y3 0

4 1

7 2

12 3

19 4

28 5

39 6

B5

31




B7

B8

: y-2 -3

-1 -2

0 1

1 6

2 13

B9

B10

Thefunctioniscontinuousandgrows

byanequalfactorof5overequal

intervals.They-interceptis(0,1).

32




1.5 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


SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP6–Attendtoprecision

Launch(WholeClass):

Besurethattheentireclassunderstandstheinstructions.Emphasizethatitisimportantforthe

discussionthattheyhavefullyjustifiedtheirchoicesontherecordingsheet.Asktheclassforideas

abouthowtheycanjustifythattwofunctionsareinversesandrecordtheirresponses.Tell

studentstoreferencethelistandtobeveryspecificaboutthefunctionsthattheyareworkingwith

astheywritetheirjustifications.Fulljustificationshouldincludethetypeoffunctionsthatare

shownintherepresentationsandhowtheycanshowthateitherthe(#, 2)valuesareswitchedover




theentiredomainofeachofthefunctionsorthatthetwofunctionsundoeachother(by

composition—eitherformallyorinformallybydescribingtheoperationsinthenecessaryorder).


Monitorstudentsastheywork,listeningforparticularlychallenging,controversial,orinsightful

commentsgeneratedbythework.Besurethatstudentsarerecordingtheirjustificationsandthey

aresufficientlyspecifictothecardstheyhaveselected.Whilelisteningtostudents,identifyone

pairofstudentstopresentforeachfunctionanditsinverse.


Iftimepermits,youmaychoosetohave10differentpairsofstudentseachdemonstrateapairof

functionsandhowtheyjustifiedthattheywereinverses.Lookforvarietyintheirjustifications,

includingcheckingtoseeifthegraphsarereflections,verifyingbycompositionofspecificvalues,or

showinghowtheinputsandoutputshavebeenswitched.Ifthereisnotenoughtimetogothrough

eachpair,thenselectthecardsthatcausedthemostdiscussionduringtheexplorationphase.The

functionpairsare:

A1 B7

A2 B3

A3 B6

A4 B8

A5 B2

A6 B1

A7 B4

A8 B5

A9 B10

A10 B9





1.5


READY Topic:PropertiesofexponentsUsepropertiesofexponentstosimplifythefollowing.Writeyouranswersinexponentialformwithpositiveexponents.1. !"

#∙ !%#

2. !& ∙ !' ∙ !(

3. )( ∙ )"

&∙ *%+

4. 32+ ∙ 9 ∙ 27&

5. 8' ∙ 16& ∙ 2(

6. 5" %

7. 7" 45

8. 346 47

9. 78'

79

%

SET Topic:RepresentationsofinversefunctionsWritetheinverseofthegivenfunctioninthesameformatasthegivenfunction.Functionf(x)

Inverse:45 !

10.x f(x)

-8 0

-4 3

0 6

4 9

8 12

10.


33




1.5


11. 12.

12.:(!) = −2! + 4

13.: ! = BCD%!

14.

15.x : !

0 0

1 1

2 4

3 9

4 16

15.

34




1.5


GO Topic:CompositefunctionsCalculateE F G HIJF E G foreachpairoffunctions.

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

16.: ! = 3! + 7; D ! = −4! − 11

17.: ! = −4! + 60; D ! = −5

6! + 15

18.: ! = 10! − 5; D ! ="

7! + 3

19.: ! = −"

%! + 4; D ! = −

%

"! + 6

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

inversesofeachother.Whichonesdoyouthinktheyare?

Why?

35

Functions & Their Inverses - mathematicsvisionproject.org€¦ · The Mathematics Vision Project...

Documents

Transcript of Functions & Their Inverses - mathematicsvisionproject.org€¦ · The Mathematics Vision Project...