Properties Of Modulus

A function is a relation between two sets in which every element of one set has only one such image in another set. One of the important functions in mathematics is the Modulus function. We study the modulus and properties of modulus in Class 11 Chapter-2 Relations & Functions.

Modulus or Absolute value function is a wonder of a function that always gives out a positive outcome, irrespective of the nature of the input! We denote modulus as |x| where x is a real number and || is the sign of modulus. Properties of modulus help it functions similar to the square root of the square of a variable. So, mathematically, |x|=x2 So, let us dive deeper into the  Properties of modulus study material.

Modulus Function

• A modulus function, also known as the Absolute Value of function, is a real-valued function that always gives out a positive or absolute value, even if the real variable is negative.
• The function(f), in which R tends to be a real number, is denoted as

f(x)={x,x0} or {-x,x<0}; where x is the component of a non-empty set.

Value of Modulus Function

The output of the modulus function if

• x is positive or zero: f(x)=x
• x is negative: f(x)=-x

Let’s consider an example

If x=-2, then f(x)=-(-2), we can also denote it as |-2|

Now since x has a negative value, we do negative of the actual value, then f(x)=-(-2) =2 

Modulus Function & the Number Line

We can measure the distance between two numbers on a number line using the Modulus function.

|x-0| =|x|, which denotes the difference between x and the origin. Similarly, the distance between two real numbers, x, and y, is |x-y|.

Example:

The distance from -3 to -5 is:

|-3-(-5)|=|-3+5|

=|2|=2

Domain & Range for Modulus Function

The domain is the set of first elements of the ordered pairs in a relation R from one set A to another set. While the range is the set of all second elements for the ordered pair for the same.

• Here, domain= R, where R stands for real numbers.
• Range= R+which is all non-negative real numbers.

Graphical Representation of Modulus Function

Let’s plot the graph for the below values of x and |x|:

Observations:

• The graph is linear and continuous starting from zero. We can thus interpret that the graph is differentiable except at zero value.
• The modulus function is an even function as the graph is symmetric about the y-axis.
• The pair of x values are the same as y when non-zero, indicating the images and preimages are not uniquely related.
• Modulus can not be used in an inverted manner.

Properties of the Modulus

• Square Function

|x|=x2

• Modulus as Equality

One of the key properties of modulus is that we can use it to represent intervals. Suppose for a non-negative number 2  we equate it to an independent variable x

|x|=2

Thus, values that satisfy this equation:

x=2

We can also observe from the modulus graph that the y=|x| and y=2 intersect, while, when y=-2, there is no intersection.

• Non-Negativity

For |f(x)|=g

• g>0

Thus, f(x)=g

• g=0

then, f(x)=0

• g<0

 No solution as modulus output can’t be negative.

• Inequality

Inequality is one of the crucial properties of modulus and depends on the type of number we choose

• If g>0

Then, if|x|0

-g<x<g

Wherein, the less than-inequality lies between the -g and g interval.

If |x|>g and g>0

Thus, x<-g or x>g 

Thus, we can extend this expression as:

|f(x)|<g; g>0

Thus, -g<f(x)<g

• If g<0

|x|

Other Properties of Modulus

1.The two variables are equal if the modulus of the difference of the two variables is zero.

If |x-y|=0,

Then x=y

2. The modulus of the sum of two real variables is less than the sum of the modulus of each variable.

|x+y||x|+|y|

3. The modulus of the difference of two real variables is more than or equal to the modulus of difference of the modulus of each variable.

|x-y|||x|-|y||

4. The modulus of the product of two variables is the product of the modulus of each variable.

|xy|=|x|*|y|

5. The modulus of the quotient of each variable is the quotient of the modulus of each real non-zero variable.

XY=|X||Y|, where |y|0

6. |x-y|<|x-z|+|z-y|, when there are three real variables x, y and z

Here, |x-y| represent the distance of x from y, |x-z| distance of x from z and |z-y| represents the difference of z from x.

Since the sum of two sides of a triangle is always more than the third, similar is valid here to express one of the crucial properties of modulus.

Conclusion

Modulus function gives an absolute value for any input variable, thus also known as Absolute value function. The value of f(x) with a modulus function will have a positive outcome. Every non-negative value of x gives f(x)=x while a negative variable’s modulus gives -x as the outcome. We can use the modulus function to find the distance of a variable from a reference point.

The graphical plot of modulus is continuous and symmetric. We found the range and domain of modulus functions as R and R+, respectively. In the end, we discussed important properties of modulus including equality, inequality, square function, etc. Now, as you have gone through the study material notes on properties of modulus, it’s time to brush up on your concepts and try out some FAQs for properties of modulus.