Question

For each pair of functions f, g below, find f(g(x)) and g(f(x))

Then, determine whether and are inverses of each other.
Simplify your answers as much as possible.
(Assume that your expressions are defined for all in the domain of the composition.
You do not have to indicate the domain.)

Answer 1

See below

Explanation:

Part A

`f(g(x))=f((x)/(3))=3((x)/(3))=x\\g(f(x))=g(3x)=(3x)/(3)=x`

Since BOTH
`f(g(x))=x` and
`g(f(x))=x`, then
`f` and
`g` are inverses of each other

Part B

`f(g(x))=f((x+1)/(2))=2((x+1)/(2))+1=x+1+1=x+2\\g(f(x))=g(2x+1)=((2x+1)+1)/(2)=(2x+2)/(2)=x+1`

Since BOTH
`f(g(x))\\eq x` and
`g(f(x))\\eq x`, then
`f` and
`g` are NOT inverses of each other