Let $M = \{(s\sin(t), s\cos(t), s^2+t)|s,t\in \mathbb R)\} \subseteq \mathbb R^3$. An obvious parametrization for $M$ would be $r:\mathbb R^2 \to \mathbb R^3, r(t,s) = (s\sin(t), s\cos(t), s^2+t)$. In order to show it is indeed a parametrization I need to show it's a homemorphism, so I need to find its inverse and prove it's also continuous. So we could write:
$$x = s\sin(t), y=s\cos(t),z= s^2+t $$
And so $t = z - s^2 = z - x^2-y^2$, and $s^2 = x^2 + y^2$. Now (and I'm not sure about this) we could divide $M$ to where $s$ is positive and where its negative, and we would get two parametrizations with two diffrent inverse functions. But apparently the correct expression of the inverse of $r$ is: $r^{-1}(x,y,z) = (x\sin(z-x^2-y^2) + y\cos(z-x^2-y^2), z-x^2+y^2)$. Looking at this formula I can see why it's the inverse of $r$, but I'm not sure how I was supposed to come up with that myself. Would appreciate any help.