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

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.’ 

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.