11.2Combining and inverting functions A-level考试纯数学P1课件.ppt

CombiningandinvertingfunctionChapter11

ReversingfunctionsThefunctions‘add2’and‘subtract2’aresaidtobeinversefunctionsofeachother.Thesefunctionssuchas‘changesign’、‘reciprocal’aresaidtobeself-inverseTheinverseofafunctionfisdenotedbythesymbolf-1

rangeoff-1domainoffxyf-1domainoffrangeoffxyf

ExampleFindthevalueofcos-1when(a)y=0.5(b)y=-1(c)y=1.5

ExampleIfand,evaluate(a)gf(50),(b)gf(130)Theanswertopart(b)callsforamorecarefullookatthetheoryofinversefunctions.

One-onefunctionsAfunctionfdefinedforsomedomainDisone-oneif,foreachnumberyintherangeRoffthereisonlyonenumberx∈Dsuchthaty=f(x).ThefunctionwithdomainRdefinedby,istheinversefunctionoff.

f-1f(x)=x,andff-1(y)=yarecalledidentityfunctionsbecausetheirinputsandoutputsareidentical.Thereisasubtledifferencebetweenthesetwocompositefunctions.ThefirsthasdomainDandthesecondhasdomainR.

Findinginversefunctionshasinverse

ExampleRestrictthedomainofthefunctionsothataninversefunctionexists.Findanexpressionforf-1y=x2–2xxy

ExampleFindtheinverseofthefunction,whereandx≠2

Graphinginversefunctionsy=f(x)DRxyy=f(x)DRxyy=f-1(x)

Iffisone-onefunction,thegraphsofy=f(x)andy=f-1(x)arereflectionsofeachotherintheliney=xExampleForthefunction.Sothataninversefunctionexists.Findanexpressionforf-1.