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!