I was reading about Riemann Sphere and I found out that from the system: $$ u(x,y) = x/(x^2+y^2+1) \\ v(x,y) = y/(x^2+y^2+1) \\ w(x,y) = (x^2+y^2)/(x^2+y^2+1) \\$$
we can find inverses: $$ x(u,v,w) = u/(1-w)\\ y(u,v,w) = v/(1-w)\\ $$ Which means that from $$ f:R^2 -> R^3$$ we have inverse function $$F: R^3-> R^2$$ Here it is easy to find these functions, but is there a general theorem that shows us when can we find inverse function from $R^n$ to $R^m$ and how to find it?
In general, inverting a rational mapping from $\mathbb R^n$ to a subset of $\mathbb R^m$, even when it is possible, is not easy. You might try Gröbner basis techniques.