How can I prove the surjectivity of the following function $F:\mathbb{R}^2 \to \mathbb{R}^2,\\  f(x,y)= ((x^3)-x),y)$
How can I check if this is surjective or not?

$y$  has no influence to $x^3-x$ and we can choose $y$ any number. 
It is enough to show that $r(x)=x^3-x$ is surjective. You can show it in many ways;
$lim_{x\to \infty}r(x)= \infty$ and $lim_{x\to -\infty}=-\infty$ and $r(x)$ continious so for any $a\in \mathbb R$ there exist $x$ such that $x^3-x=a$
