How to determine the domain of definition of a function $y(x)$ defined implicitly:
Hint: There is no general method to obtain the domain of definition (I assume you mean maximal domain of definition) for implicit functions. But for your examples, there is an easy method.
Solve the equations for $x$. The image of the function $x=x(y)$ is the maximal domain of definition for the function $y=y(x)$.