Change of variable pdf inverse function

Question

I've been given the following problem: $f(x,y) = e^{-(x+y)}$ on intervals $x \ge 0$ and $y \ge 0$, and $f(x,y) = 0$ otherwise. I'm also given that $Φ_1(x,y) = \frac{x}{y} = U$ and $Φ_2(x,y) = x + y = V$. I've proven that $f(x,y)$ is a pdf as asked by the problem but then the problem asks me to find the inverse functions of $Ψ_1(U,V) = x$ and $Ψ_2(U,V) = y$. I don't have a clue how to go about this: I certainly know how to find the inverse of a function but this looks like nothing I've seen in my textbook thus far.

Answer 1

If x/y = u and x+y = v then x = uy hence uy+y = v, thus y = v/(u+1) and x = uv/(u+1). That is, Ψ1(u,v) = uv/(u+1) and Ψ2(u,v) = v/(u+1).