# 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!