Surjection/Examples/Real Square Function to Non-Negative Reals

Example of Surjection
Let $f: \R \to \R_{\ge 0}$ be the real square function whose codomain is the set of non-negative reals:
 * $\forall x \in \R: \map f x = x^2$

Then $f$ is a surjection.

Proof
Let $y \in \R_{\ge 0}$.

Let $x = +\sqrt y$.

From Existence of Square Roots of Positive Real Number, there exists such a $y$.

Then:
 * $x^2 = y$

That is:
 * $y = \map f x$

Hence:
 * $\forall y \in \R_{\ge 0}: \exists x \in \R: y = \map f x$

and the result follows by definition of surjection.