Answer
The domain of the given function is
$$
\left\{(x, y) | x \ne -y \right\}.
$$
Work Step by Step
$$
g(x,y)=\frac{x-y}{x+y}
$$
This function is defined only when:
$$
x+y \ne 0
$$
$\Rightarrow$
$$
x \ne -y
$$
Therefore, the domain of the given function is
$$
\left\{(x, y) | x \ne -y \right\}.
$$