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Modulus Function
A modulus function gives the magnitude of a number irrespective of its sign. It is also called the absolute value function. In mathematics, the modulus of a real number x is given by the modulus function, denoted by x. It gives the nonnegative value of x. The modulus or absolute value of a number is also considered as the distance of the number from the origin or zero.
In this article, we will learn about the modulus function definition, calculating modulus for numbers, variables, and polynomials along with solved examples and modulus function questions.
What is Modulus Function?
The modulus function, which is also called the absolute value function gives the magnitude or absolute value of a number irrespective of the number being positive or negative. It always gives a nonnegative value of any number or variable. Modulus function is denoted as y = x or f(x) = x, where f: R → (0,∞) and x ∈ R.
x is the modulus of x, where x is a real number. If x is nonnegative then f(x) will be of the same value x. If x is negative, then f(x) will be the magnitude of x, that is, f(x) = x if x is negative. Let us sum up the modulus function formula below.
Modulus Function Formula
The value of the modulus function is always nonnegative. If f(x) is a modulus function, then we have:
 If x is positive, then f(x) = x
 If x = 0, then f(x) = 0
 If x < 0, then f(x) = x
This means if the value of x is greater than or equal to 0, then the modulus function takes the actual value, but if x is less than 0 then the function takes minus of the actual value 'x'.
Domain and Range of Modulus Function
We can apply the modulus function to any real number. The range of the modulus function is the set of nonnegative real numbers which is denoted as (0,∞) and the domain of the modulus function is R (where R refers to the set of all real numbers). Hence, the domain of the modulus function is R and the range is (0,∞).
Application of Modulus Function
Now, that we know the modulus function formula, let us consider a few examples to understand its application. The steps to calculate modulus function are given below:
Example: Consider the modulus function f(x) = x.
 If x = − 3, then y = f(x) = f(−3) = −(−3) = 3, here x is less than 0.
 If x = 3, then y = f(x) = f(3) = 3, here x is greater than 0
 If x = 0, then y = f(x) = f(0) = 0, here x is equal to 0
Note that f(3) = f(3) here. In other words, 3 = 3 = 3.
Graph of Modulus Function
Now let us see how to plot the graph for a modulus function. Let us consider x to be a variable, taking values from 5 to 5. Calculating modulus for the positive values of 'x', the line plotted in the graph is 'y = x' and for the negative values of 'x', the line plotted in the graph is 'y = x'.
x  f(x) = x 

5  5 
4  4 
3  3 
2  2 
1  1 
0  0 
1  1 
2  2 
3  3 
4  4 
5  5 
Properties of Modulus Function
Now, that we have the formula for the modulus function and the graph of the modulus function, let us now explore the properties of the modulus function:
 Property 1: The modulus function always gives a nonnegative number as output for all real values of x. Thus, the modulus function is never equal to a negative number.
x = a; a > 0 ⇒ x = ± a ;
x = a; a = 0 ⇒ x = 0 ;
If x = a, then a can never be less than zero.  Property 2:
 Case 1: (If a > 0): Inequality for a positive number
f(x) < a and a > 0 ⇒ −a < f(x) < a
f(x) > a and a > 0 ⇒ −a < f(x) (or) f(x) > a  Case 2: (If a < 0): Inequality for a negative number
f(x) < a and a < 0 ⇒ there is no solution for this.
f(x) > a and a < 0 ⇒ this is valid for all real values of f(x).
 Case 1: (If a > 0): Inequality for a positive number
 Property 3: If x,y are real numbers, then
 x = x
 x − y = 0 ⇔ x = y
 x + y ≤ x + y
 x − y ≥ x − y
 xy = x y
 x/y = x/y, y not equal to zero.
Derivative And Integral of Modulus Function
Since we know that a modulus function f(x) = x is equal to x if x > 0 and x if x < 0, therefore the derivative of modulus function is 1 if x > 0 and 1 if x < 0. The derivative of the modulus function is not defined for x = 0. Hence the derivative of modulus function can be written as d(x)/dx = x/x, for all values of x and x not equal to 0.
Using the formula of the modulus function and integration formulas, the integral of the modulus function is (1/2)x^{2} + C if x ≥ 0, and its integral is (1/2)x^{2} + C if x < 0. Hence the integration of the modulus function can be clubbed as:
 ∫x dx = (1/2)x^{2} + C if x ≥ 0
 ∫x dx = (1/2)x^{2} + C if x < 0
Important Notes on Modulus Function
 The modulus function is also called the absolute value function and it represents the absolute value of a number. It is denoted by x.
 The domain of modulus functions is the set of all real numbers.
 The range of modulus functions is the set of all real numbers greater than or equal to 0.
 The vertex of the modulus graph y = x is (0,0).
 The vertex of the modulus function y = a x  h + k is (h, k).
Topic Related to Modulus Function
Modulus Function Examples

Example 1: Find the value of the modulus function x for x = 5 and x = 10
Solution:
If x = 5, then x = 5 = 5
If x = 10, then x = 10 = 10
Answer: x = 5 for x = 5 and x = 10 for x = 10

Example 2: Solve x + 3 = 8 using modulus function definition.
Solution:
We know that the modulus function value is always nonnegative, therefore we have two cases:
If x + 3 > 0, then x + 3 = x + 3 and if x + 3 < 0, then x + 3 = (x + 3).
Case 1: If x + 3 > 0, we have
x + 3 = x + 3
⇒ x + 3 = 8
⇒ x = 8  3 = 5
Case 2: If x + 3 < 0, we have
x + 3 = (x + 3)
⇒ (x + 3) = 8
⇒ x  3 = 8
⇒ x = 3  8 = 11
Answer: Hence, the solution is x = 5, 11

Example 3: Solve the inequality x  1 < 3.
Solution:
Here, 3 > 0. So by the properties of modulus function,
3 < x  1 < 3
Adding 1 on all the sides,
3 + 1 < x < 3 + 1
2 < x < 4
Answer: The solution of the given inequality is 2 < x < 4.
FAQs on Modulus Function
What is Modulus Function?
The modulus function gives the magnitude or absolute value of a number irrespective of the number being positive or negative. Modulus function is denoted as y = x or f(x) = x, where f: R → R and x ∈ R. This is also called the absolute value function.
What is the Domain and Range of the Modulus Function?
The domain of the modulus function is R (where R refers to the set of all positive real numbers) and the range of the modulus function is the set of nonnegative real numbers which is denoted as (0,∞).
What is the Integration of Modulus Function?
The integration of the modulus function depends on the value of x. It is:
 ∫x dx = (1/2)x^{2} + C if x ≥ 0
 ∫x dx = (1/2)x^{2} + C if x < 0
How to Differentiate Modulus Function?
We have f(x) = x is equal to x if x > 0 and x if x < 0, hence, the derivative of modulus function is 1 if x > 0 and 1 if x < 0. The derivative of the modulus function is not defined for x = 0. To summarize, the derivative of modulus function x is x/x, where x not equal to 0.
Is Modulus Function Always Positive?
The modulus of a positive number is positive. The modulus of a negative number is obtained by ignoring the minus sign. Thus, the modulus function is always positive.
Why Modulus Function is Not Differentiable?
Modulus function x is not differentiable at x = 0 as the graph of Mod(x) has a sharp point at x = 0. Also, the lefthand limit and the righthand limit of the derivative are not equal at x = 0.
How do You Solve Modulus Function Problems?
Modulus functions problems can be solved by applying modulus to a nonnegative number and a negative number always results in the same number.
How do you Graph Modulus Function?
Take some positive and negative values of x. Also, take x = 0. Frame a table with two columns x and y with all the random xvalues that we have chosen. Calculate the modulus of every number and fill in the column of y. Then just plot all the ordered pairs (x, y) and join them by lines. We will get a 'V'  shaped graph and it is the graph of the modulus function.
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