precalc episode iv
TRANSCRIPT
Functions:EpisodeIVBytheendofthislesson,Iwillbeabletoanswer
thefollowingquestions…1.HowdoIperformarithmeticcombinationsoffunctionsandhowaretheyarerepresentedgraphically?
2.HowdoIbuildcompositefunctionsanddeterminetheirdomain?
3.HowdoIbuildaninversefunctionalgebraicallyfromanoriginalfunction?4.Whatarethecharacteristicofinversefunctions?5.Whatisaone-to-onefunction?
6.Inversefunctionnotation:Iff(x)andg(x)areinverses,g(x)canberenamed
7.One-to-onefunction:Whentheinverseofafunctionisafunctionalso.
f −1 x( )
Vocabulary
PrerequisiteSkillswithPracticeCalculatorexerciseintroducingthe
storagebuttonandthevariablebutton.Putthefollowingequationsintermsofx:
y = 2x − 45x +1
y = − (x − 3)3
2+10
UnderstandingFunctionNotationGiventhefollowingfunctions,
performtheindicatedoperation.f (x) = 2x −1g(x) = 6x2 + x − 2
h(x) = x
f (3)− g(−2)= gf
⎛⎝⎜
⎞⎠⎟(x)=
(g! f )(x)=3h(16x4 )= 2g(t2 −1)=
Compositionoffunctions:Pluggingfunctionsintootherfunctions.f (x) = x2 −1g(x) = 2x −1
Giventhefunctionsonthetheleft,findandThenevaluatethefunctionsat1,2&3yourgraphingcalculator.
( f ! g)(x)(g ! f )(x)
x
1
2
3
x
1
2
3
( f ! g)(x) (g! f )(x)
Astoneisthrownintoapond.Acircularrippleisspreadingoverthepondinsuchawaythattheradiusisincreasingattherateof5.3feetpersecond.Findafunction,r(t),fortheradiusintermsof“t”.FindaFunction,A(r),fortheareaoftherippleintermsof“r”.Find
(A ! r)(t)
Compositionoffunctions:Asimpleapplication
DomainsandCompositeFunctions
f (x) = x
g(x) = 1x
h(x) = 3x2 −10x − 8l(x) = x2 −16
(g !h)(x)Giventhefollowingfunctions,findtheDOMAINofeach.
(l ! f )(x)
(g ! g)(x) ( f ! l)(x)*ConsidertheDomainofthefunctionbeinginput.ThenconsidertheDomain
ofthesimplifiedbuild.Thetherestrictedelementsbothconditionsabovemakethefinalcomposite
domain.
UsingPropertiesofInversestoVerifyInverses
Definitionofinversefunctions.
Supposef(x)andg(x)areinversefunctions.Thefollowingwouldholdtrue….
1.f[g(x)]=xandg[f(x)]=x
2.TheDomainoff(x)becomestheRangeofg(x)andRangeoff(x)becomestheDomaing(x)
3.Graphsoff(x)andg(x)reflectaboutthey=xaxis.
Verifythatandareinverses.
f (x) = 2x3 −1 g(x) = x +12
3
1.Switchxandy.2.Solvefory.
Otherthingstoconsider…• One-to-one?• Restricteddomain?• Inversecan’tbefoundbyconventionalmeans?
FindingInverseAlgebraically
f (x) = −2 3 x + 4 g(x) = x + 2 − 3
h(x) = x2
4+1 l(x) = x
x − 4+ 6
m(x) = 2x3 − x + 3CHALLA
NGE!
Interpretinginversevalues/regularvaluesfromagraph.
f −1(2)=g−1(−1)=f ! g( )(−1)=f −1 ! f( )(3)=( f ! g)(−2)=
f x( )
g x( )