Image of $A \subset \mathbb{R}^2$ under general transformation

Question

If I have a transformation $\varphi: \mathbb{R}^2 \to \mathbb{R}^2$ which doesn't have any particular property, for example, which is not linear, how do I know what is the image of a subset $A \subset \mathbb{R}^2$ under this transformation?

I know how to compute this if $\varphi$ is linear and invertible, but is there a method if $\varphi$ is a general application? What if $\varphi(x,y)=(x^2,0)$ ? What is the image of $\{(x,y): x^2+y^2=1\}$ under this transformation?

I'm not asking for a trick for this particular application, I'm looking for a general method. I've been thinking about this but couldn't figure it out.

Thank you!

Answer 1

Your question is very very very very very broad. In general, given a set $A\subseteq X$ and a set $B\subseteq X$, as long as $A$ has at least as many elements as $B$, you can find some function $\phi: X\to X$ such that $\phi(A)=B$.

The function may not be pretty (for example, it may not be continuous), but it does exist.

Answer 2

For your given $A$ we have $x=\sqrt{1-y^2}$ and $y$ ranges from $[-1,1]$.

So $\phi (A) =\{\phi (\sqrt{1-y^2}, y) | y \in [-1,1]\}$ = $\{(1-y^2,0)|y \in [-1,1]\}$

However, in general it is not something that one can simply answer other than saying $\phi (A) = \{\phi (a) | a \in A\} $