$Iff\left(x\right)=px+qandg\left(x\right)=rx+sandf\left[g\left(x\right)\right]=g\left[f\left(x\right)\right]\iff$

$\begin{array}{l}f\left(x\right)=px+q,g\left(x\right)=rx+s\\ Givenf\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)\\ \Rightarrow f(rx+s)=g(px+q)\\ \Rightarrow p(rx+s)+q=r(px+q)+s\\ \Rightarrow prx+ps+q=prx+rq+s\\ \Rightarrow ps+q=rq+s\\ f\left(s\right)=g\left(q\right)\end{array}$