What is the domain of the function $f(x,y)= \sqrt{xy} $.

In this case, domain is $D=\{(x,y) \in R^2: \}$ 

Since $ \sqrt{xy} = x^{0.5}y^{0.5} $ I can write the function equivalently as $f(x,y)= x^{0.5}y^{0.5} $ but in this case domain is only $D=\{(x,y) \in R^2: x \geq 0, y \geq0 \}$ 

So, which one is the domain finally? Is it possible that the function $f$ has different domain depending on how you write it? This is confusing.