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.
Modulus Function Graph
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.
Similar Reads
Function in Maths
A function in math is like a machine that takes an input, does something to it, and gives a specific output. For each input, there’s exactly one output. It’s a rule that connects each input to one and only one result. Functions are fundamental in fields like algebra and calculus. They help model rel
4 min read
Functions in LISP
A function is a set of statements that takes some input, performs some tasks, and produces the result. Through functions, we can split up a huge task into many smaller functions. They also help in avoiding the repetition of code as we can call the same function for different inputs. Defining Functio
3 min read
Types of Functions
Functions are defined as the relations which give a particular output for a particular input value. A function has a domain and codomain (range). f(x) usually denotes a function where x is the input of the function. In general, a function is written as y = f(x). A function is a relation between two
15 min read
Signum Function
Signum Function is an important function in mathematics that helps us to know the sign of a real number. It is usually expressed as a function of a variable and denoted either by f(x) or by sgn(x). It may also be written as a sign(x). Signum Function also has applications in various fields such as p
5 min read
Many One Functions
Many-to-One or Many One Function is one of the various types of functions that represent relationships between different entities. As we know, a function is a very specific type of relation in which each input has a unique output. In a many-one function, many inputs can have the same output. For exa
7 min read
Solidity - Function Modifiers
Function behavior can be changed using function modifiers. Function modifier can be used to automatically check the condition prior to executing the function. These can be created for many different use cases. Function modifier can be executed before or after the function executes its code. The modi
8 min read
Math Functions in PL/SQL
In PL/SQL, mathematical functions play a important role in performing calculations and manipulating numeric data. These functions allow us to execute a wide range of mathematical operations from basic arithmetic to complex computations within our PL/SQL code. In this article, we will learn about Mat
4 min read
Less.js Math mod() Function
Less.js is a simple CSS pre-processor that facilitates the creation of manageable, customizable, and reusable style sheets for websites. It is chosen because CSS is a dynamic style sheet language. LESS is flexible, thus it functions with a variety of browsers. Web browsers can only use CSS that has
3 min read
Piecewise Function
Piecewise Function is a function that behaves differently for different types of input. As we know a function is a mathematical object which associates each input with exactly one output. For example: If a function takes on any input and gives the output as 3. It can be represented mathematically as
11 min read
Less.js Math Functions
In this article, we will see the various Math functions that are provided by Less.js to perform various mathematical functions as per the user's requirements within CSS code only. Less.js (Leaner Style Sheets) is an extension to normal CSS which basically enhances the abilities of normal CSS and giv
6 min read