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Modulus function

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The modulus function is both a fascinating and crucial topic in mathematics for competitive exams.The modulus function always returns a positive result for any variable or an integer .Because it gives a non-negative result for every independent variable, whether positive or negative, it is also known as the absolute value function. 

For any variable or integer, the modulus function always returns a positive value.The absolute value function is another name for it.The modulus of a real number x is given by the modulus function, represented as |x| in mathematics.It returns x as a non-negative value. The modulus, or absolute value, of a number is frequently referred to as the distance from the origin, or zero. 

Definition :

The modulus function, also known as the absolute value of a function, determines the magnitude or absolute value of a number, regardless of whether it is positive or negative.

Any number or variable will always have a non-negative value. 

y = |x| or f(x) = |x| denotes the modulus function, where f: R→R and x   R. 

The modulus of x, where x is a real number, is called |x|. If x is not negative, f(x) has the same value as x. If x is negative, f(x) will be the magnitude of x, i.e. f(x) = -x. 

Modulus Function Formula:

The modulus function always has a positive value. If f(x) is a modulus function, we get the following:

If x is positive, then f(x) = x 02

If x = 0, then f(x) = 0

If x < 0, then f(x) = -x

 

                                      

This means that the modulus function takes the actual value if x is more than or equal to 0, but if x is less than 0, the function takes the negative of the actual value ‘x’.  

Graph of Modulus Function:

Let’s look at how to draw a graph for a modulus function. Consider x as a variable with values ranging from -5 to 5. When calculating modulus, the line plotted in the graph for positive values of ‘x’ is ‘y = x,’ while for 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


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Properties of Modulus Function:

Now that we have the modulus function’s formula and graph, let’s look at the modulus function’s properties: 

  • Property 1: For all real values of x, the modulus function returns a non-negative number. It’s also incorrect to convert the modulus function to a negative number. 

x = a |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

Inequality of a negative number

|f(x)| < a and a > 0 −a < f(x) < a

Case 2: (If a < 0)

 |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).

  • Property 3: If x and y are actual numbers, 

|-x| = |x|

|x − y| = 0 x = y

|x + y| ≤ |x| + |y|

|x − y| ≥ ||x| − |y||

|xy| = |x| |y|

x/y| = |x|/|y|, where y is not zero. 

Important Notes on Modulus Function:

  • The set of all real numbers greater than or equal to 0 is the range of modulus functions.

  • The set of all real numbers is the domain of modulus functions.

  • The y= |x| vertex of the modulus graph is (0,0).

  • The absolute value function, commonly known as the modulus function, represents the absolute value of an integer.|x| is the symbol for it. 

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Frequently asked questions

Get answers to the most common queries related to the JEE Examination Preparation.

What is the Modulus Function's Domain and Range?

Ans.  The domain of the modulus function is R (where R is the set of all posi...Read full

Is Modulus Function Always Positive?

Ans: A positive number’s modulus is positive.By disregarding the minus s...Read full

What exactly is the Modulus Function?

Ans:  The modulus function, also known as the absolute value of a function, determines the magnitude or absolute va...Read full

For x= -5 and x=10, find the value of the modulus function |x|.

Ans: If x = -5, then |x| = |-5| = 5 If x = 10, then |x| = |10| = 10...Read full

What is the best way to graph the modulus function?

Ans: We can build the graph of the modulus function by using negative values like...Read full