Inverse functions mcTYinverse Aninversefunctionisasecondfunctionwhichundoesthewor koftherstone

Inverse functions mcTYinverse Aninversefunctionisasecondfunctionwhichundoesthewor koftherstone - Description

Inthisunit wedescribetwomethodsfor64257ndinginversefunctionsandwe alsoexplainthatthedomainofa functionmayneedtoberestrictedbeforeaninversefunctio ncanexist Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercises ID: 25145 Download Pdf

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Inverse functions mcTYinverse Aninversefunctionisasecondfunctionwhichundoesthewor koftherstone

Inthisunit wedescribetwomethodsfor64257ndinginversefunctionsandwe alsoexplainthatthedomainofa functionmayneedtoberestrictedbeforeaninversefunctio ncanexist Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercises

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Inverse functions mcTYinverse Aninversefunctionisasecondfunctionwhichundoesthewor koftherstone




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Inverse functions mc-TY-inverse-2009-1 Aninversefunctionisasecondfunctionwhichundoesthewor kofthefirstone. Inthisunit wedescribetwomethodsforfindinginversefunctions,andwe alsoexplainthatthedomainofa functionmayneedtoberestrictedbeforeaninversefunctio ncanexist. Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature. Afterreadingthistext,and/orviewingthevideotutorialo nthistopic,youshouldbeableto: understandthedifferencebetweeninversefunctionsandrec iprocalfunctions,

findaninversefunctionbyreversingtheoperationsapplied to intheoriginalfunction, findaninversefunctionbyalgebraicmanipulation, understandhowtorestrictthedomainofafunctionsothatit canhaveaninversefunction, sketchthegraphofaninversefunctionusingthegraphofthe originalfunction. Contents 1. Introduction 2. Workingout byreversingtheoperationsof 3. Usingalgebraicmanipulationtoworkoutinversefunctions 4. Restrictingdomains 5. Thegraphof www.mathcentre.ac.uk 1 math centre2009
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1. Introduction Supposewehaveafunction thattakes to ,sothat ) = y. Aninversefunction,whichwecall

,isanotherfunctionthattakes backto .So ) = x. For tobeaninverseof ,thisneedstoworkforevery that actsupon. Key Point Theinverseofthefunction isthefunctionthatsendseach backto . Wedenotethe inverseof by 2. Working out by reversing the operations of Onewaytoworkoutaninversefunctionistoreversetheopera tionsthat carriesoutona number. Hereisasimpleexample. Weshallset ) = 4 ,sothat takesanumber and multipliesitby4: ) = 4 (multiplyby4). Wewanttodefineafunctionthatwilltake4times ,andsenditbackto .Thisisthesame assayingthat divides by4.So ) = (divideby4).

Thereisanimportantpointaboutnotationhere.Youshouldn oticethat doesnotmean /f .Forthisexample, /f wouldbe withthe inthedenominator,andthatisnot thesameas Hereisaslightlymorecomplicatedexample.Supposewehave ) = 3 + 2 Wecanbreakupthisfunctionintoaseriesofoperations.Fir stthefunctionmultipliesby3,and thenitaddson2. + 2 + 2 www.mathcentre.ac.uk 2 math centre2009
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Togetbackto from ,wewouldneedtoreversetheseoperations. Sowewouldneedt takeaway2,andthendivideby3.Whenweundotheoperations, wehavetoreversetheorder aswell. 2) x x + 2 + 2 Nowwehavereversedalltheoperationscarriedoutby

,andsoweareleftwith ) = Hereisonemoreexampleofhowwecanreversetheoperationso fafunctiontofinditsinverse. Supposewehave ) = 7 Itiseasiertoseethesequenceofoperationstobecarriedou ton ifwerewritethefunctionas ) = + 7 Sothefirstoperationperformedby takes to ;thentheresultismultipliedby ;and finally7isaddedon. (cube) 1) + 7 + 7 Sotogetfrom to ,weneedtostartbytakingaway7.Thenweneedtoundotheoper ation ‘multiplyby ’,sowedivideby .Andfinallyweundothefirstoperationbytakingthecube root. (cube) (cuberoot) 1) 1) + 7 + 7 Nowwehavereversedeveryoperationcarriedoutby .So ) = x.

www.mathcentre.ac.uk 3 math centre2009
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Key Point Wecanworkout byreversingtheoperationsof .Ifthereismorethanoneoperationthen wemustreversetheorderaswellasreversingtheindividual operations. Exercises 1.Workouttheinversesofthefollowingfunctions: (a) ) = 6 , (b) ) = 3 + 4 , (c) ) = 1 3. Using algebraic manipulation to work out inverse functions Anotherwaytoworkoutinversefunctionsisbyusingalgebra icmanipulation. Wecandemon- stratethisusingoursecondexample, ) = 3 + 2 Nowtheinversefunctiontakesusfrom backto .Ifweset ) = 3 + 2 then isthefunctionthattakes to .Sotoworkout

weneedtoknowhowtogetto from .Ifwerearrangetheexpressionfor weobtain = 3 + 2 2 = 3 sothat Sowewant ) = ( 2) ,andthisisexactlythesameassayingthatthefunction is givenby ) = ( 2) Wecanusethemethodofalgebraicmanipulationtoworkoutin verseswhenwehaveslightly trickierfunctionsthantheoneswehaveseensofar.Letusta ke ) = , x > Wehavemadetherestriction x > becauseat = 1 thefunctiondoesnothaveavalue.This isbecausethedenominatoriszerowhen = 1 www.mathcentre.ac.uk 4 math centre2009
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Nowweset x/ 1) .Multiplyingbothsidesby weget 1) = x, andthenmultiplyingoutthebracketgives yx x.

Wewanttorearrangethisequationsothatwecanexpress asafunctionof ,andtodothis wetakethetermsinvolving totheleft-handside,giving yx y . Nowwecanthentakeout asafactorontheleft-handsidetoget 1) = y , anddividingthroughoutby wefinallyobtain Sotheinversefunctionis ) = y/ 1) ,andthisisexactlythesameassayingthatthe function isgivenby ) = x/ 1) .Sointhiscase happenstobethesameas Inthelastexample,itwouldnothavebeenpossibletoworkou ttheinversefunctionbytryingto reversetheoperationsof .Thisexampleshowshowusefulitistohavealgebraicmanipu lation toworkoutinverses. Key Point

Algebraicmanipulationisanothermethodthatcanbeusedto workoutinversefunctions. Exercises 2.Usealgebraicmanipulationtoworkout foreachofthefollowingfunctions: (a) ) = for x > , (b) ) = + 1 + 2 for x > www.mathcentre.ac.uk 5 math centre2009
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4. Restricting domains Notallfunctionshaveinverses.Forexample,letusseewhat happensifwetrytofindaninverse for ) = ? ? ) = Whenwedefineaninversefunctionfor ,welookforanotherfunctionthattakesthevalues andgivesusback .Butinthecaseof ) = therearetwovaluesof thatgivethe same . Thisisbecauseboth ) = andalso ) = . Wecannotdefine of

somethingtobetwodifferentthings. Togetaroundthisproblem,werestrictthedomainofthefunc tion.Soforexamplewith ) = ,ifwedefinethefunctiononlyfor thenthegraphlookslikethis. ? ) = 0 Sonowwehaveexactlyonevalueof givingeachvalueof .Thisrestrictedversionof canhaveaninverse.Theinverseis ) = + www.mathcentre.ac.uk 6 math centre2009
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Wecouldinsteadhaverestricted to . Thisstillgivesonlyonevalueof foreach valueof .Thistimetheinverseis x. ? ) = 0 Hereisanotherexampleofafunctionthatweneedtorestrict inordertodefineaninverse.Let uslookat ) = sin

.Thegraphofthefunctionlookslikethis. ) = sin x Thistimeifwetrytodefinetheinversefunction,weseethatt herearemanypossiblevaluesof foreach . Weneedtorestrictthedomainofourfunctionsothatwearel ookingata sectionwithonlyasinglevalueof foreachvalueof .Wedothisbysettingthedomainof sin tobe 90 90 www.mathcentre.ac.uk 7 math centre2009
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) = sin x 90 90 Itisparticularlyimportantheretorememberthat sin istheinversefunctionto sin ,and thatitdoesnotmean (sin ,eventhoughthesimilarexpression sin doesmean (sin Forthisreason,theinversefunction sin issometimescalled arcsin

Itisalsopossibletodefinetheinversefunctions cos and tan byrestrictingthedomains ofthefunctions cos and tan . Theseinversefunctionsarealsocalled arccos and arctan andyoucanfindoutmoreaboutthemintheunitonTrigonometri cFunctions. Somefunctionscannothaveinverses,evenifwerestrictthe irdomains.Forexample,aconstant functioncannothaveaninverse. ) = 4 ? Howeversmallwemakethedomain,therearealwayslotsofval uesof givingthesamevalueof .Theonlywaywecangetasinglevalueof isbyrestrictingthedomaintoasinglepoint. Sowesaythatthisfunctionhasnoinverse. Exercises

3.Whichofthesefunctionsneedtohavetheirdomainsrestri ctedinordertodefineaninverse? Howwouldyourestricttheirdomains? (a) ) = + 2 + 1 , (b) ) = , (c) ) = 5 , (d) ) = sin 2 www.mathcentre.ac.uk 8 math centre2009
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5. The graph of Thereisaneasywaytoworkoutthegraphofaninversefunctio ,usingthegraphofthe originalfunction Supposethatwehavethegraphofsomefunction .Thenapointonthegraphof willhave co-ordinates x, f )) , )) Rememberthataninversefunctionsends backto . Sothegraphof mustcontain thepoints , x .Soifweinterchangethe and axes,wewillgetthegraphoftheinverse function. , )) ),

Bygoingfromthegraphof tothegraphof ,wearereflectingthegraphinthediagonal line.Butthisdiagonallineistheline .Sothegraphof isjustthegraphof reflected intheline Exercises 4.Foreachofthefollowing,sketchthegraphof anduseittosketchthegraphof (a) ) = 3 , (b) ) = 2 + 5 , (c) ) = for (d) ) = 1 /x for x > , (e) ) = www.mathcentre.ac.uk 9 math centre2009
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Answers 1. (a) ) = (b) ) = (c) ) = 2. (a) ) = 3 + 4 (b) ) = 3. (a) or (b) norestrictionneeded (c) or (d) 45 45 4. (a) (b) (c) (d) (e) www.mathcentre.ac.uk 10 math centre2009