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The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education
This work is licensed under the Creative Commons Attribution CC BY 4.0
MODULE 1
Functions & Their Inverses
SECONDARY
MATH THREE
An Integrated Approach
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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
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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.
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
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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
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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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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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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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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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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.2
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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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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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10.Considerthefunction!(#) = 0.03#)overthedomainofallrealnumbers,notjustthedomainofthisproblemsituation.Howdoesthegraphchangefromthegraphofd(s)inquestion#3?
11.Howdoeschangingthedomainofd(s)changethegraphoftheinverseofd(s)?
12.Istheinverseofd(s)afunction?Justifyyouranswer.
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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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SECONDARY MATH III // MODULE 1
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?
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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.
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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
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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.
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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( )
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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
CC B
Y Ch
ristia
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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:
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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
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:
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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
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
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SECONDARY MATH III // MODULE 1
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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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SECONDARY MATH III // MODULE 1
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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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SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
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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.
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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
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:____________________________________
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_sam
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http
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uUq2
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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:____________________________________
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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
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
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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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.
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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
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
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.5
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B7
B8
: y-2 -3
-1 -2
0 1
1 6
2 13
B9
B10
Thefunctioniscontinuousandgrows
byanequalfactorof5overequal
intervals.They-interceptis(0,1).
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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
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11. 12.
12.:(!) = −2! + 4
13.: ! = BCD%!
14.
15.x : !
0 0
1 1
2 4
3 9
4 16
15.
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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