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$F:\mathbb{R}^2 \to \mathbb{R}^2,\\ f(x,y)= ((x^3)-x),y)$

How can I check if this is surjective or not?

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Have you looked up the definition of surjective? –  gebruiker May 4 at 14:02
    
Try to balance the parentheses. –  Marc van Leeuwen May 4 at 17:32

2 Answers 2

up vote 3 down vote accepted

This is surjective since $\forall\alpha\in\Bbb R:x^3-x=\alpha$ has at least one real root.

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$y$ 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 $x^3-x=a$

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