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 –modulus function

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!

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