Modulus Function, Properties of Modulus Function - Practice Questions & MCQ

Modulus Function:

The function f: RR defined by f(x) = |x| for each x R is called the modulus function.

For each non-negative value of x, f(x) is equal to x. But for negative values of x, the value of f(x) is the negative of the value of x

$|\mathrm{x}|, \quad \mathrm{x} \in \mathbb{R}=\left\{\begin{array}{cc}x, & x \geq 0 \\ -x, & x<0\end{array}\right.$

Range $\in[0, \infty)$

Modulus Equations: Properties

If $a>0$
1. $|x|=a$, then $x=a,-a$
2. $|x|=|-x|$
3. $|x|^2=x^2$
4. If $|\mathrm{x}|=\mathrm{x}$, then $\mathrm{x}>0$ or $\mathrm{x}=0$
5. If $|x|=-x$, then $x<0$ or $x=0$
6. $|f(x)|=|g(x)|$, then $f(x)=g(x)$ or $f(x)=-g(x)$

Modulus inequalities

These deal with the inequalities (<, >, ≤, ≥ ) on expressions with absolute value sign. 

Properties

If a, b > 0, then

1.
$
\begin{aligned}
& |x| \leq a \Rightarrow x^2 \leq a^2 \\
& \Rightarrow-a \leq x \leq a
\end{aligned}
$
2.
$
\begin{aligned}
& |x| \geq a \Rightarrow x^2 \geq a^2 \\
& \Rightarrow x \leq-a \text { or } x \geq a
\end{aligned}
$
3.
$
\begin{aligned}
& a \leq|x| \leq b \Rightarrow a^2 \leq x^2 \leq b^2 \\
& \Rightarrow x \in[-b,-a] \cup[a, b]
\end{aligned}
$
4. $|x+y|=|x|+|y| \Leftrightarrow x y \geq 0$.
5. $|x-y|=|x|-|y| \Rightarrow x \cdot y \geq 0$ and $|x| \geq|y|$
6. $|x \pm y| \leq|x|+|y|$
7. $|x \pm y| \geq||x|-|y||$