Modulus functions

Understanding the modulus function

Numbers can be positive or negative (or zero). We sometimes will like to convert our numbers into positive numbers (non-negative to be exact). The modulus function, denoted by $| \cdot |,$ does that.

For example $| 3 | = 3,$ $| 0 | = 0$ and $| - 5 | = 5 .$

This is sometimes also called the absolute value function.

Modulus as a piecewise function

The modulus function can be defined as a piecewise function:

$| f (x) | = {\begin{matrix} f (x) & if f (x) \geq 0 \\ - f (x) & if f (x) < 0 \end{matrix}$

Example

For example, consider the function $f (x) = | x - 2 | .$

If $x < 2,$ then $x - 2$ is negative so the function can be rewritten as $f (x) = - (x - 2) = 2 - x .$

On the other hand, if $x \geq 2,$ then $x - 2$ is non-negative so the function can be rewritten as $f (x) = x - 2 .$

Finding inverses of modulus functions

Question:

$\begin{aligned} f : x \mapsto | x - 2 |, & x < 1, \\ g : x \mapsto | x - 2 |, & x \geq 2 . \end{aligned}$

Find $f^{- 1} (x)$ and $g^{- 1} (x) .$

Solution:

When $x < 1,$ then $x - 2$ is negative so $f (x) = - (x - 2)$

$\begin{aligned} Let y & = - (x - 2) \\ y & = - x + 2 \\ x & = 2 - y \end{aligned}$

$f^{- 1} (x) = 2 - x ∎$

When $x \geq 2,$ then $x - 2$ is non-negative so $g (x) = x - 2$

$\begin{aligned} Let y & = x - 2 \\ x & = y + 2 \end{aligned}$

$g^{- 1} (x) = x + 2 ∎$