Image of Real Square Function

Theorem
The image of the real square function is the entire set of positive real numbers $\R_{\ge 0}$.

Proof
From Square of Real Number is Non-Negative, the image of $f$ is $\R_{\ge 0}$.

From Positive Real has Real Square Root:
 * $\forall x \in \R: \exists y \in \R: x^2 = y$

Hence the result by definition of image.