Absolute Value is Many-to-One

Theorem
Let $f: \R \to \R$ be the absolute value function:

\forall x \in \R: f \left({x}\right) = \begin{cases} x & : x \ge 0 \\ -x & : x < 0 \end{cases} $

Then $f$ is a functional relation.

Proof
Let $f \left({x_1}\right) = y_1, f \left({x_2}\right) = y_2$ where $y_1 \ne y_2$.

The result follows.