What is the Domain and Range of Modulus Function

Here, you will learn modulus function and what is the domain and range of modulus function.

Let’s begin –

The function f(x) defined by

y = |x| = $\begin{cases} x & \text{if}\ x \ge 0 \\ -x & \text{if}\ x < 0 \end{cases}$

is called the modulus function.

It is also called absolute value function.

we observe that the domain of the modulus function is the set R of all real numbers and the range is the set of all non-negative real numbers. 

Domain and Range of Modulus Function

For f(x) = |x|,

Domain is R

Range is [0,$\infty$]

The graph of the modulus function is as shown in figure, for x > 0, the graph coincides with the graph of the identity function i.e. the line y = x and for x > 0, it is coincident to the line y = -x, 

The modulus function has the following properties :

(i) For any real number x, we have

$\sqrt{x^2}$ = |x|

For example, $\sqrt{cos^2x}$  = | cos x |  = $\begin{cases} cos x ,  & 0 \ge x \le \pi/2 \\ -cos x , & \pi/2 < x \le \pi \end{cases}$

(ii) If a, b are positive real numbers, then

$x^2$ $\le$ $a^2$ $\iff$ |x| $\le$ a $\iff$ -a $\le$ x $\le$ a

$x^2$ $\ge$ $a^2$ $\iff$ |x| $\ge$ a $\iff$ x $\le$ -a or, x $\ge$ a

$x^2$ < $a^2$ $\iff$ |x| < a $\iff$ -a < x < a

$x^2$ > $a^2$ $\iff$ |x| > a $\iff$  x < -a or, x > a

$a^2$ $\le$ $x^2$ $\le$ $b^2$ $\iff$ a $\le$ |x| $\le$ b $\iff$ x $\in$ [-b, -a] $\cup$ [a, b]

$a^2$ < $x^2$ < $b^2$ $\iff$ a < |x| < b $\iff$ x $\in$ (-b, -a) $\cup$ (a, b)

(iii) For any real number x and y, we have

| x + y | = | x | + | y |, if (x $\ge$ 0 and y $\ge$ 0) or, (x < 0 and y < 0)

| x – y | = | x | – | y |, if (x $\ge$ 0 and | x | $\ge$ | y |) or, (x $\ge$ 0 and y $\le$ 0 and | x | $\ge$ | y |)

| x $\pm$ y | $\le$ | x | + | y |

| x $\pm$ y | > | | x | – | y | |

Hope you learnt what is the domain and range of modulus function, learn more concepts of function and practice more questions to get ahead in the competition. Good luck!