I would like to determine a parametrization of the set $S = \{(x,y,z) \in \mathbb{R}^3 \; \vert \; x^2 + y - z^2 = 1\}$ as a graph of some smooth function $h: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}$.
As it should be the graph, I am seeking for something of the form $\sigma(u,v) = (u,v, h(u,v))$. I thought I could simply solve according to $z$, thus $z^2 = x^2 + y - 1 \equiv z = \pm \sqrt{x^2+y-1}$, but at this point I would get two solutions for $z$. I am a little bit confused, how to construct the smooth function $h$. I would appreciate suggestions.
In order to avoid square roots, I think it is better to consider something like that $$\mathbb{R}^2\ni (x,z)\to\sigma(x,z) = (x,z^2-x^2+1,z).$$ The surface is a hyperbolic paraboloid.