I apologize in advance if this is a duplicate (I would expect to see something like this on here), but I haven't found anything that matches my question.
I have this general difficulty (perhaps, to lack of practical experience) of finding and writing down images of functions given a domain. For example, I have recently seen this problem in a book:
Given $f(x,y) = \left(\begin{array}{c}u(x,y)\\v(x,y)\end{array}\right) = \left(\begin{array}{c}x^2-y^2\\2xy\end{array}\right)$, find image of $f$ when $x>0$, $y>0$, $x^2+y^2<1$
Basically, it's asking for a set of inequalities for what $u$ and $v$ trace out in the $u$-$v$ plane. How do I go about solving such problems for a general case of $f : \mathbb{R}^m \to \mathbb{R}^n$? For example, in the above problem, the domain of $f$ is a quarter of a circle of radius 1 in the first quadrant of the $x$-$y$ plane. How do I find what $f$ maps this to in the $u$-$v$ plane? Where do I start?
Update: So apparently (by empirically constructing a table of values), $f$ maps the quarter of a circle to a half-circle ($v$>0, $u^2+v^2<1$). Is there a way to see this intuitively? Most importantly, how can I get these two inequalities algebraically?