Real Square Function is not Surjective

Example of Mapping which is Not a Surjection
Let $f: \R \to R$ be the real function defined as:
 * $\forall x \in \R: \map f x = x^2$

Then $f$ is not a surjection.

Proof
For $f$ to be a surjection, it would be necessary that:
 * $\forall y \in \R: \exists x \in \R: \map f x = y$

However from Square of Real Number is Non-Negative:
 * $\forall y \in \R_{< 0}: \nexists x \in \R: \map f x = y$

Hence $f$ is not a surjection.