If two random variables $X$ and $Y$ are functions of the same one random variable $Z$, how do we find the joint PDF of $X$ and $Y$? I have seen examples where two RVs are functions of the other two RVs, but what if they are function of the same RV? Do we just "make up" a second RV, say, $V$ such that it is independent from $Z$ and has some simple distribution (it shouldn't matter I guess which one) and then proceed as in the "normal" case? But what would the inverse functions for $Z$ and $V$ be in this case? We need them for the Jacobian, right?
Also, what if there is no explicit inverse for functions $f$? For example, say $X = Z + \exp(Z)$, how do we find a PDF of $X$?