Answer
The domain of the given function is
$$
\left\{(x, y) | x^{2}+y^{2} \ne 1, \quad x \lt 2 \right\}.
$$
Work Step by Step
$$
g(x,y)=\frac{\ln (2-x)}{1-x^{2}-y^{2}}
$$
This function is defined only when:
$$
1-x^{2}-y^{2} \ne 0
$$
$\Rightarrow$
$$
x^{2}+y^{2} \ne 1
$$
In addition, $g$ is not defined if:
$$
2-x \gt 0
$$
$\Rightarrow$
$$
x \lt 2
$$
Thus, the domain of $g $ is
$$
\left\{(x, y) | x^{2}+y^{2} \ne 1, \quad x \lt 2 \right\}.
$$
