Answer
Domain of the given function is
$$
\left\{(x, y) | x^{2} \leq y , \quad x\ne \pm1 \right\}.
$$
Work Step by Step
$$
g(x,y)=\frac{\sqrt {y-x^{2}}}{1-x^{2}}
$$
This function is defined only when:
$$
1-x^{2} \ne 0
$$
$\Rightarrow$
$$
x^{2} \ne 1, \quad i.e. (x\ne 1, x\ne -1)
$$
In addition, $g$ is not defined if:
$$
y-x^{2} \geq 0
$$
$\Rightarrow$
$$
x^{2} \leq y
$$
Thus the domain of $g $ is
$$
\left\{(x, y) | x^{2} \leq y , \quad x\ne \pm1 \right\}.
$$
