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I was thinking about finding the domain of multivariable function below,

$$f(x,y)=(-1)^{x+y}$$

But I am not sure what approach I should take to find answer to such questions. How should find the domain of such function?

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    Well, what x and y values can you plug in or not plug in ? – 2017-02-13
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    Consider what x+y can take on. – 2017-02-13
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    @VivekKaushik Well, I know that and I have checked some powers of $-1$ we get but the problem is I couldn't find a satisfactory answer to it. – 2017-02-13

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This depends on whether you are working in $\mathbb R$ or $\mathbb C$.

In $\mathbb R$, one notes that for integer $n$,

$$(-1)^{2n+1} = -1 \implies -1=(-1)^{1/(2n+1)}$$

and further, for integer $m$,

$$(-1)^m = \boxed{(-1)^{m/(2n+1)}} = -1\text{ or }+1$$

We cannot generalize the exponent to be all of rational numbers, since $(-1)^{1/(2n)}$ is not real.

In $\mathbb C$:

$$(-1)^a := e^{a\log(-1)} = e^{a(i\pi+2\pi i n)}$$

for integer $n$. This is defined for all $a\in \mathbb C$.