Conjugate function under linear transformation

Question

I am learning about nonlinear transformation and I just saw a problem on a book, it asks for the formula for the conjugate function of $g(x)=f(Ax)$, where $A$ is an $m\times n$ matrix, $f:R^m\to R, g:R^n\to R$. So, I know the formula for the situation that $m=n$ and $A$ is invertible, that is very easy:By definition of conjugate function, we have:$$g^*(s)=\sup_{x}(<s,x>-g(x))=\sup_{x}(<s,x>-f(Ax))$$Substitute $Ax$ by $z$, we have:$$g^*(s)=\sup_{z}(<s,A^{-1}z>-f(z))=\sup_{z}(<A^{-T}s,z>-f(z))=f^*(A^{-T}s)$$But I have no idea how things would be if $A$ is a general matrix, I thought about using pseudo-inverse, it would be like $$g^*(s)=f^*((A^+)^Ts)$$But this just seems so strange, I feel like something is wrong here, but I don't know why, and I wonder what a correct answer should look like. Could somebody help me? Thank you so much.