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):

$g(x)=\frac{x+3}{4}$

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:

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

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