Modulus Function

1 Modulus Function

The modulus function (also known as the absolute value function) \(\lvert \cdot \rvert:\mathbb{R}\to \mathbb{R}^+\) is defined by \[ \lvert x \rvert = \begin{cases}x & \text{if }x\geq 0 \\ -x & \text{if } x<0.\end{cases} \]

Example: Let use define the linear function \(f(x)=4x+7\). The plot of \(\lvert f(x) \rvert\) is:

1.1 Properties of the Modulus Function

Consider real number \(a>0\). Then: 1. \(\lvert f(x) \rvert<a \implies-a<f(x)<a\); 2. \(\lvert f(x) \rvert>a \implies f(x)<-a~\text{or}~f(x)>a\).

For real numbers \(x,y\in \mathbb{R}\) we have that 1. \(\lvert -x \rvert=\lvert x \rvert\), 2. \(\lvert x-y \rvert=0 \iff x=y\), 3. \(\lvert x+y \rvert\leq \lvert x \rvert+ \lvert y \rvert\), 4. \(\lvert x-y \rvert\geq \lvert \lvert x \rvert -\lvert y \rvert\rvert\), 5. \(\lvert xy \rvert=\lvert x \rvert\lvert y \rvert\), 6. \(\left\lvert \frac{x}{y} \right\rvert= \frac{{\lvert x \rvert}}{\lvert y \rvert}\).