# Absolute Value Function is Even Function

## Theorem

Let $\size {\, \cdot \,} : \R \to \R$ denote the absolute value function on $\R$:

Then $\size {\, \cdot \,}$ is an even function.

## Proof

Recall the definition of the absolute value function:

$\size x = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$

Testing the $3$ cases in turn:

 $\displaystyle x$ $>$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle -x$ $<$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \size {-x}$ $=$ $\displaystyle x$ Definition of Absolute Value $\displaystyle$ $=$ $\displaystyle \size x$ Definition of Absolute Value

$\Box$

 $\displaystyle x$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle -x$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \size {-x}$ $=$ $\displaystyle 0$ Definition of Absolute Value $\displaystyle$ $=$ $\displaystyle \size x$ Definition of Absolute Value

$\Box$

 $\displaystyle x$ $<$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle -x$ $>$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \size {-x}$ $=$ $\displaystyle -x$ Definition of Absolute Value: as $-x > 0$ $\displaystyle$ $=$ $\displaystyle \size x$ Definition of Absolute Value: as $x < 0$

$\Box$

Hence the result by definition of even function.

$\blacksquare$