Answer
The Domain is
$$(x,y) \in (0,\infty)\times \mathbb{R}.$$
The Range is
$$g(x,y) \in (-\infty,\infty) = \mathbb{R}.$$
Work Step by Step
1) Domain.
The expression under the square root has to be nonnegative so we demand that $x\geq0$. But since $\sqrt{x}$ is in the denominator it must not be zero so we need that $x>0$ because the square root of a strictly positive number is strictly positive. $y$ is in the numerator so it can take any real value. So for the domain we have
$$x\in(0,\infty),\quad y\in (-\infty,\infty) =\mathbb{R};$$
$$(x,y) \in (0,\infty)\times \mathbb{R}.$$
2) Range
This is the set of all possible values of $y$. Since $y$ can take any real value and it is divided by a strictly positive number always that we see that $g(x,y)$ can take any real value so for the range we have
$$g(x,y) \in (-\infty,\infty) = \mathbb{R}.$$
