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