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.’
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 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.