Answer
The Domain is given by
$$x\in \mathbb{R}/\{0\},\quad y\in \mathbb{R}/\{0\}.$$
The Range is given by
$$z(x,y)\in \mathbb{R}/\{0\}.$$
Work Step by Step
1) Domain
Here the only restriction is that the denominator $xy$ must not be zero. This means that both $x$ and $y$ must not be zero. All other combinations of values for $x$ and $y$ are allowed so the domain is given by
$$x\in \mathbb{R}/\{0\},\quad y\in \mathbb{R}/\{0\}.$$
2) Range
Since both $x$ and $y$ cannot be zero at the same time then $x+y$ is also different than zero always. Since we divide the numerator which takes all the real values except zero by the denominator that also takes all the real values except zero then the value for $z$ also takes all the real values except zero, so the range is given by
$$z(x,y)\in \mathbb{R}/\{0\}.$$
