Question

Given g(x)=(x+3)/4, find g^{-1}(x). If necessary, indicate any domain restrictions.Prove that it is the inverse through composition.

Answer 1

Given the function g(x):

`g(x)=(x+3)/(4)`

First, we determine the inverse of g(x)

`\begin{gathered} 4g(x)=x+3\implies4y=x+3 \\ x=4y-3 \\ \text{Therefore:} \\ g^(-1)(x)=4x-3 \end{gathered}`

Next, to verify if they are inverses by composition, we check if the following holds:

`g(g^(-1)(x))=g^(-1)(g(x))=x`

This is done below:

`\begin{gathered} g(g^(-1)(x))=(\lbrack4x-3\rbrack+3)/(4)=(4x)/(4)=x \\ g^(-1)(g(x))=4\lbrack(x+3)/(4)\rbrack-3=x+3-3=x \end{gathered}`

Therefore, we have shown that they are inverses by composition.