How to find domain of a complex function?
For example, say a function $f(z) = z^2$ maps a region $D$ of complex plane to a region $R$ which is a disc of radius $4$ and centre $4 + 0 i$, then, how can I find the region $D$?
My background: I am not familiar with the properties of complex function and calculus relevant to that, so I can't attempt that.
Though, I tried following things:
I took some simple number like $4 + 0i , 2 + 0i , 8 + 0i$ and applied $f^{-1} (z)$ , I found the domain $D$ has not circular boundary because the scaling factor for any number $z$ is not fixed.
I thought it is something parametrised region like in multivariable calculus, but couldn't come with something good, because I can't account for the rotation factor which is in complex number geometry.
$D=\{z:z^{2}\in R\}=\{z:|z^{2}-4|<4$. Writing real and imaginary parts of $z$ as $x$ and $y$ we have have to find all $z$ such that $|x^{2}-y^{2}+2ixy-4|^{2} <16$ which reduces to $ (x^{2}-y^{2}-4)^{2}+4x^{2}y^{2} <16$ or $(x^{2}-y^{2})^{2}-8(x^{2}-y^{2})+4x^{2}y^{2}<0$. You can write this as $ (x^{2}+y^{2})^{2} <8(x^{2}-y^{2})$. I don't think this can be further simplified.