Answer
Domain of the given function is
$$
\left\{(x, y) | x \geq 3y \right\}.
$$
Work Step by Step
$$
f(x,y)=\sqrt[4] {x-3y}
$$
This function is defined only when:
$$
x-3y\geq 0
$$
$\Rightarrow$
$$
x\geq 3y
$$
Thus the domain of $f $ is
$$
\left\{(x, y) | x \geq 3y \right\}.
$$