Answer
The domain of the given function is
$$
\left\{(x, y) | x^{2}+y^{2} \geq 4 \right\}.
$$
Work Step by Step
$$
f(x,y)= \sqrt {x^{2}+y^{2}-4}
$$
This function is defined only when:
$$
x^{2}+y^{2}-4 \geq 0
$$
$\Rightarrow$
$$
x^{2}+y^{2} \geq 4
$$
Therefore, the domain of the given function is
$$
\left\{(x, y) | x^{2}+y^{2} \geq 4 \right\}.
$$
