1.4 Pulling a Rabbit Out of the Hat1.4 Pulling a Rabbit Out of the Hat A Solidify Understanding Task I have a magic trick for you: • Pick a number, any number. • Add 6 • Multiply

SECONDARY MATH III // MODULE 1

FUNCTIONS AND THEIR INVERSES – 1.4

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

Ihaveamagictrickforyou:

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

Peopleareoftenmystifiedbysuchtricksbutthoseofuswhohavestudiedinverseoperationsandinversefunctionscaneasilyfigureouthowtheyworkandevencreateourownnumbertricks.Let’sgetstartedbyfiguringouthowinversefunctionsworktogether.

Foreachofthefollowingfunctionmachines,decidewhatfunctioncanbeusedtomaketheoutputthesameastheinputnumber.Describetheoperationinwordsandthenwriteitsymbolically.

Here’sanexample:

Input Output

!(#) = # + 8 !)*(#) = # − 8

# = 7 7 7 + 8 = 15

Inwords:Subtract8fromtheresult

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20

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FUNCTIONS AND THEIR INVERSES – 1.4

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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

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FUNCTIONS AND THEIR INVERSES – 1.4

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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

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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

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

A Solidify Understanding Task

Purpose:Thepurposeofthistaskistosolidifystudents’understandingoftherelationshipbetween

functionsandtheirinversesandtoformalizewritinginversefunctions.Inthetask,studentsare

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:

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• 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:

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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.

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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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1.4

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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

24

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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

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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