Properties of Modulus Function

A modulus function determines a number’s magnitude regardless of its sign. The absolute value function is another name for it. The domain of the modulus function is all real numbers, and the range is all positive real numbers and 0. In this article, we will learn about the modulus function, its meaning, application, and properties through video lessons. The objective of these notes is to help you prepare for the competitive examinations. 

Meaning of Modulus Function

According to modulus function, a modulus function is a function that returns a number or variable’s absolute value. It generates the number of variables’ magnitudes. A number’s modulus, or absolute value, is sometimes known as the number’s distance from the origin or zero. The modulus function gives the modulus of a real number x, represented as |x| in mathematics. It returns a non-negative value for x, implying that the function’s output is always positive, regardless of the input.

The function is denoted as y = |x| or f(x) = |x|, 

where f: R → R and x ∈ R.

f(x)= x         if x0   or   f(x)= -x       if x ＜0

Thus, |x| is the modulus of x, where x is a real number. The modulus function takes the actual value of x if x is more than or equal to 0. However, if x is less than 0, the function takes the minus of the actual value ‘x’.

Application of Modulus Function

The application of the modulus function is relatively easy to understand, and you can learn through video lessons. The modulus function always gives the absolute value of an input x as the result of the function. 

To understand the application of the formula, let us consider a few solved examples:

Solved Example: Consider the modulus function f(x) = |x|. Then:

• If x = -7, then y = f(x) = – (-7) = 7, since x is less than zero
• If x = 5, then y = f(x) = 5, since x is greater than zero
• If x = 0, then y = f(x) = 0, since x is equal to zero

Thus keeping the above mentioned rules in mind, we can solve any modulus function. 

Domain and Range of Modulus Function

The domain of a function refers to the range of values that can be put into it. This is the set of x values in a function like f(x). On the other hand, a function’s range is the set of values that the function can take. This is the set of values that the function produces when we enter an x value.

The domain of the modulus function is R, where R represents the set of all positive real numbers. At the same time, its range is the set of non-negative real numbers, denoted as [0,∞) and R+. Any real number can be modulated using the modulus function. 

Therefore, it can be said that the range of the modulus function is (0,∞) and the domain is R. 

Properties of Modulus Function

The properties of a modulus function can be iterated as follows:

For any a, b belonging to R:

Property 1: |a| ≥ 0

Property 2: For all real values of x, the modulus function returns a positive number. It is also incorrect to convert the modulus function to a negative number.

Thus,

|x| = a; a > 0 ⇒ x = ± a;
|x| = a; a = 0 ⇒ x = 0;
If |x| = a, then the value of a can not be less than zero.

Property 3: (If a > 0)

Inequality of a negative number
|f(x)| < a and a > 0 ⇒ −a < f(x) < a

Thus, for example:

|f(x)| < 2

Here, 2 > 0

Therefore, it is true that −2 < f(x) < 2
Inequality for a positive number
|f(x)| > a and a > 0 ⇒ f(x) > a or f(x)<-a

Thus, for example:

|f(x)| > 3

Here, 3 > 0

Therefore, it is true that  f(x) > 3 or f(x)<-3

Property 4: (If a < 0)
|f(x)| < a and a < 0; This function does not have a possible solution.

|f(x)| > a and a < 0; This means this function holds true for all real values of f(x).

Property 5: If a, b are real numbers, then

• |-a| = |a|

For example: |-3| = |3|

• |a − b| = 0 ⇔ a = b

For example: |2 – 2| = 0; this means a = b

• |a + b| ≤ |a| + |b|

For example: |7 + 3| = |7| + |3|

|5 + (-3)| < |5| + |-3|

• |a − b| ≥ ||a| − |b||

For example: |8 – 2| = | |8| – |2| |

|8 – (-2)| > | |8| – |-2| |

• |ab| = |a| |b|

For example: |(4)(5)| = |4| |5|

|(-4)(5)| = |-4| |5|

• |ab| = |a||b|, where y is not equal to zero.

For example: |-82| = |-8||2|

Property 6

• x2 ≤ a2 which means |x| ≤ a; Thus -a ≤ x ≤ a
• x2 ≥ a2 which means |x| ≥ a; Thus -a ≥ x ≥ a
• x2 < a2 which means |x| < a; Thus -a < x < a
• x2 > a2 which means |x| > a; Thus -a > x > a

Conclusion

Thus we have studied the meaning, application and various properties of the modulus function. A modulus function is a function that returns a number or variable’s absolute value and is represented by |x|. It generates the number of variables’ magnitudes. The domain of the modulus function is all real numbers, and the range is all positive real numbers and 0. Hopefully, these notes will help you in preparation.