Real Square Function is not Surjective

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Example of Mapping which is Not a Surjection

Let $f: \R \to \R$ be the real square function:

$\forall x \in \R: \map f x = x^2$

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.

$\blacksquare$