How to show that $\{(x, y) \in \mathbb{R}^2\mid\cos(x^2) + x^3 - 4 7y > e^x - y^2\}$ is open or closed? Show that the set of points $(x, y) \in \mathbb{R}^2$  such that $\cos(x^2) + x^3  - 4 7y > e^x - y^2$  is an open subset of $\mathbb{R}^2$.
I was reading the fact that if $(X,d)$ and $(Y,d)$ are metric spaces and $f:(X,d) \to \rightarrow (Y,d)$ is continuous and if $v$ be a open set (or closed set) in $Y$ then $f^{-1}(v)$ is open (closed) in $X$. I don't really under stand how to do the applications for this lemma. So far I have proved using this theorem that the identity function and the constant function are continuous. I want help to understand this problem and for similar problems I want to know what I am looking for here.what are the steps to show that a given set is closed or open using this lemma.

$f(x,y)=\cos (x^{2})+x^{3}-47y-e^{x}+y^{2}$ defines  a continuous function and the given set is $f^{-1} (0,\infty)$. 
