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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

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

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Claritathoughtabouthowmuchfoodtheycouldbuyforagivenamountofmoneyanddrewthisgraph:

2.WritetheequationofthefunctionthatClaritagraphed.

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

4. Whatistherelationshipbetweenthetwofunctions?Howdoyouknow?

5.Usefunctionnotationtowritetherelationshipbetweenthefunctions.

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

6. ModeltherelationshipbetweenthepriceandtheamountoffoodusingCarlos’approach.

7. ModeltherelationshipbetweenthepriceandtheamountoffoodusingClarita’sapproach.

8. Whatistherelationshipbetweenthesetwofunctions?Howdoyouknow?

9. Usefunctionnotationtowritetherelationshipbetweenthefunctions.

10. WhichcompanyshouldClaritaandCarlosbuytheirBrutusBitesfrom?Why?

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1.1 Brutus Bites Back – Teacher Notes A Develop Understanding Task

Purpose:Thepurposeofthistaskistodevelopandextendtheconceptofinversefunctionsina

linearcontext.Inthetask,studentsusetables,graphs,andequationstorepresentinversefunctions

astwodifferentwaysofmodelingthesamesituation.Therepresentationsexposetheideathatthe

domainofthefunctionistherangeoftheinverse(andviceversa)forsuitablyrestricteddomains.

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

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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?

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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( )

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