Homeomorphism between region above parabola and $ \mathbb R ^2$ 
Define the set $ X = \{ (x,y) \in \mathbb R ^2 : x^2 < y \}.  $
  Construct a homeomorphism between $ X $ and $ \mathbb R ^2 $.

Graphically, $ X $ is the region above the parabola $y = x^2$, not including the boundary lines. Since it's contained entirely in the upper half plane, I thought of trying something involving logarithms and considered $\Phi: X \to \mathbb R ^2, \; (x,y) \to (x, \ln \sqrt y)$. As it stands, $f$ is injective and also continuous as the composition of continuous maps on the domain of definition but I wasn't able to show surjectivity. For any  $ (x,y) \in \mathbb R ^2 $ we have $(x,y) = \Phi (x, e^{2y} ) $, but the latter isn't necessarily contained in $X$ so I'm not sure how to proceed (or even if this kind of function is the right form to be considering).

Hint: For the strategy $(x,y)\mapsto(x,\star)$ to work, you'll want a function of both $x$ and $y$ in the $\star$.
