**TI83F* AppVariable file 12/11/02, 11:50=+ ,+STUFUNC,+*+G5HVilg/ ]DJ "#&'Study of FunctionsDonna RobertsOctober 2002, 2008V2Study of FunctionsStudyofFunctions(Hit1)  &H?0 8    0    ?    < <       PThefollowingsetofstudycardsdealswithfunctions,domains,range,continuous,odd/even,&inverse,algebraand-composition.P Vertical line test*Downarrowforgraph*1.Graphisafunction2.GraphisNOTafunction3.Cannotbedetermined-P 8T@p`` p ``p`p``p`0p ` `@`@p@@` p ` `0p`p`D"D&b"2A/pxd s`p``p`p``p``p`p``p`pPThisgraphpassestheverticallinetestandisafunction.Thismeansthatforeveryxvalue,there&existsoneandonlyone-correspondingyvalue.P)Zero denominators| *Downarrowtochoices*Whatisthedomainofthefunction  A!D  |!D|?"D0$@P0 P<`#*1P1.allrealnumbers2.allrealsexcluding33.allrealsexcluding-34.allrealsexcludingboth3and-3P Thisfunctionhasapossiblezerodenominatorproblem.Since2x-6=0whenx=3,thevalue3must&beexcludedfromthe-domain.PFinding range*Downarrowforgraph*Whatistherangeforthisfunction?1.{y|yisanyrealnumber}2{y|-1=43.allreals>-44.allreals>=-4PForthisfunctiontobedefined,thevalueunderthesquarerootmustbegreaterthanorequalto0.Theexpressionx+4is&greaterthanorequalto0-whenxis>=-4.P Double trouble *Downarrowtochoices*Whatisthedomainofthefunction  D `   P >!D > ?!D"D@" $>#*1P1.allrealnumbers2.allreals>=23.allreals>24.allrealsexcluding2P Thisfunctionhastwoproblems-possiblezerodenominatorANDworkablesquarerootvalues.Thevaluesx>=2willsatisfythe&squareroot,butx=2will-causeazerodenominator.PContinuous*Downarrowtograph:Thisfunctionis1.continuous#2.NOTcontinuous*3.cannotbedetermined1P 8SYPIntuitivelyspeaking,agraphiscontinuousifitcanbedrawnwithoutremovingyourpencilfrom#thepaper.Thisgreatest*integergraphisnot1continuous.POdd/Even  *Downarrowtochoices*Thefunction $I  B 4@@@@?@(@H@(0 #*1P1.isodd2.iseven3.isneitheroddnorevenPODD:f(-x)=-f(x)EVEN:f(-x)=f(x)Thisfunctionisodd.  T `      p` 4|aAA(#H(p0 /XPInverse Function*Downarrowtograph* Theinverseofthisfunctionisalsoafunction.1.True&2.False-P 8Q @@  0  @ @@@`   DDL 0PHorizontallinetest: Iftheinverseofafunctionisitselfafunction,ahorizontallinemustintersecttheoriginal&functioninonlyone-location.PAddition[*Downarrowtochoices*Iff(x)=3x+2andg(x)=2x,then(f+g)(x)=-P1.3x+42.5x+23.6x+24.6x+4&PTheadditionoffunctionsissimplytheadditionoftheyvaluesatalllocationswherethedomains&intersect.-(3x+2)+(2x)=5x+2PCompositionY*Downarrowtochoices*Iff(x)=2x-1andg(x)=xthen(f(g(x))=-P1.(2x-1)2.(2x)-13.2x-14.x+2x-1PWhenworkingwithcompositionoffunctions,f(g(x)),besuretostartbyreplacingtheinnermost&parentheses.- f(g(x))=f(x)=2x-1PSplit definitions*Downarrowtochoices*Whengraphingsplitdefinitionfunctions(piecewisedefined)ontheTI-83/84+,thedomain#restriction-1=0&3.ally<=0-P 8T@`O0@`@@0`@`@`@@0`@`@@`@@`@`0@@`@`@`0AB|DDDIDHO@@`@`@@`@`@@`@@`@@`@`@@`@`P Therangeoftheabsolutevaluefunctionisallyvaluesgreaterthanorequaltozero.P