Answer
The domain is given by $$(x,y)\in (-\infty,\infty)\times (-\infty,\infty) =\mathbb{R^2}.$$
The range is the interval $(0,\infty)$.
Work Step by Step
We have that $$f(x,y) = e^{xy}$$ The exponential is defined for every real number so the product $xy$ can be any real number which means that both $x$ and $y$ can take any real value i.e. $$x\in(-\infty,\infty)=\mathbb{R},\quad y\in(-\infty,\infty)=\mathbb{R};$$ $$(x,y)\in (-\infty,\infty)\times (-\infty,\infty) =\mathbb{R^2}.$$
As for the range, the values of the exponent with the positive basis (as $e$ is) are always positive. In this case, since $xy$ will take all real values, then the value of $e^{x,y}$ will cover all positive reals i.e. the range is $(0,\infty)$.
