Modulus Function

Modulus function gives the absolute value or magnitude of a number irrespective of the number is positive or negative. The modulus function is denoted as y = |x| or f(x) = |x|, where f: R→ [0, ∞) and x ∈ R. In this article we will explore modulus function, modulus function formula domain and range of modulus function, modulus function graph, modulus function properties. We will also discuss the application of modulus function and derivative and integral of modulus function. Let's start our learning on the topic "Modulus Function".

What is Modulus Function?

Modulus function is also called as the absolute value functions as it converts number into its absolute value irrespective of whether the number is positive or negative. Modulus function is denoted as:

y = |x|

or

f(x) = |x|

Modulus Function Formula

Modulus function formula is given by:

y = |x| = \begin{array}{cc} \bigg \{ \begin{array}{cc} x & if x\geq 0 \\ -x & x<0 \end{array} \end{array}

Modulus function gives the same number if the number is positive or zero and gives negative of the number when number is negative. In other words, the modulus function result in same number when number is greater than or equal to zero and result in negative of the number when number is less than zero.

Domain and Range of Modulus Function

Below are the domain and range of the modulus function.

Domain of Modulus Function

The domain of modulus function is set of all real numbers.

Domain of modulus function |x| = R

Range of Modulus Function

The range of modulus function is all positive numbers.

Range of modulus function |x| = [0, ∞)

Application of Modulus Function

Some of the applications of modulus functions are listed below.

• Modulus function is used to measure distance on the number line.
• Modulus function is used for defining piecewise functions.
• It is also used in solving absolute value equalities and inequalities.
• Modulus function is also used in different fields like computer science, signal processing etc.

Modulus Function Graph

The below graph represents the modulus function.

Properties of Modulus Function

The properties of modulus functions are listed below:

Inequalities Property

• If a > 0, |x| < a ⇒ -a < x < a
• If a > 0, |x| > a ⇒ x ∈ (-∞, -a) ∪ (a, ∞)
• If a < 0, |x| > a is valid for all real numbers.

If a and b are Two Real Numbers

• |-a| = a
• |a - b| = 0 ⇔ a = b
• |a + b| ≤ |a| + |b|
• |a - b| ≥ |a| - |b|
• |ab| = |a| |b|
• |a / b| = |a| / |b|, b≠ 0

Derivative And Integral of Modulus Function

The derivative and integral of modulus function are discussed below.

Derivative of Modulus Function

The derivative of modulus function is given by:

(d / dx) (|x|) = x / |x|

Integral of Modulus Function

The integral of modulus function is given by:

∫ |x| dx = (1 / 2)x² + C if x ≥ 0

∫ |x| dx = -(1 / 2)x² + C if x < 0

Modulus Function Examples

Example 1: Find the value of |x| and |y| if the values of x and y are 29 and -55 respectively.

Solution:

By modulus function formula:

y = |x| = \begin{array}{cc} \bigg \{ \begin{array}{cc} x & if x\geq 0 \\ -x & x<0 \end{array} \end{array}

Value of |x| = |29| = 29

Value of |y| = | (-55) | = 55

Example 2: Determine the domain and range of modulus function p(x) = 5 - |x - 4|

Solution:

p(x) = 5 - |x - 4|

The above modulus function is defined for x ∈ R.

So, the domain of the modulus function p(x) is R i.e., set of all real numbers.

|x - 4| ≥ 0

or

- |x - 4| ≤ 0

Adding 5 both sides

5 - |x - 4| ≤ 5

So, the range of the modulus function p(x) is (-∞, 5].

Example 3: Solve: |y - 5| = 14 using modulus function definition.

Solution:

|y - 5| = 14

By definition of modulus function

y = |x| = \begin{array}{cc} \bigg \{ \begin{array}{cc} x & if x\geq 0 \\ -x & x<0 \end{array} \end{array}

If (y - 5) ≥ 0 then, (y-5)

y - 5 = 14

y = 19

If (y - 5) < 0 then, -(y - 5)

-(y - 5) = 14

-y + 5 = 14

y = -9

So, the value of y = 14 or -9.

Example 4: Solve the inequality: |q - 6| > 8

Solution:

|q - 6| > 8

Since, 8 > 0 by the property of modulus function

-8 < q - 6 < 8

Adding 6 in all sides

-8 + 6 < q < 8 + 6

-2 < q < 14

So, the solution of given inequality is -2 < q < 14.

Practice Questions on Modulus Function

Q1. Find the value of |x| and |y| if the values of x and y are 40 and -23 respectively.

Q2. Determine the domain and range of modulus function f(x) = 11 + |x - 6|

Q3. Solve: |a - 15| = 8 using modulus function definition.

Q4. Solve the inequality: |b - 11| > -5.