Finding set of functions $ f\left(u,v\right)=u^{2}+3v^{2} $
$g\left(x,y\right)=\begin{pmatrix} e^{x}cosy  \\ e^{x}siny \end{pmatrix}   $
How do I determine sets of $f\left(\mathbb R ^{2} \right)$ and set of $g\left(\mathbb R ^{2} \right)$ ?
I determine that f represent elliptic paraboloid and g represent circle, but how to write their sets?

After some comments, I think I understand what you mean so do please correct me if I'm wrong. Let's first have a definition.
If $h:\mathbb R^n\to\mathbb R^m$ is a function and $A\subset \mathbb R^n$ then
$$h(A) = \{ h(a)\in \mathbb R^m \,|\, a\in A\}$$
Then using your definitions of $f,g$ above we have:
$$f(\mathbb R^2) = \{x\in \mathbb R \,|\, x\ge0\}$$
$$g(\mathbb R^2) = \mathbb R^2 - \{(0,0)\}$$
Can you prove these?
E.g. That $f(u,v)\ge 0$ for all $u,v$ and that if $x>0$ then there is some $u,v$ such that $f(u,v)=x$
