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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
1.4 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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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
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
• 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
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.4
Needhelp?Visitwww.rsgsupport.org
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
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.4
Needhelp?Visitwww.rsgsupport.org
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 –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.4
Needhelp?Visitwww.rsgsupport.org
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