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( )