Modulus Function

Modulus Function

A modulus function gives the magnitude of a number irrespective of its sign. It is also called the absolute value function. 

In this mini-lesson we will learn about the modulus function definition, calculating modulus for numbers, variables and polynomials along with solved examples and modulus function questions.

Try out the mod function calculator to find the modulus of a number!

Lesson Plan


What Is Modulus Function?

The modulus of a function, which is also called the absolute value of a function gives the magnitude and absolute value of a number irrespective of the number being positive or negative. It always gives a non-negative value of any number or variable.

It is represented as 

\(\begin{align}y = |x|\end{align}\)

or

\(\begin{align}f(x) = |x|\end{align}\)

where \(\begin{align}f :R\rightarrow R \end{align}\) and \(\begin{align}x \in R\end{align}\)

\(\begin{align}|x|\end{align}\) is the modulus of \(\begin{align}x\end{align}\), where \(\begin{align}x\end{align}\) is a non-negative number.

If  \(\begin{align}x\end{align}\) is positive then \(\begin{align}f(x)\end{align}\) will be of the same value \(\begin{align}x\end{align}\). If \(\begin{align}x\end{align}\) is negative, then \(\begin{align}f(x)\end{align}\) will be the magnitude of \(\begin{align}x\end{align}\).

To sum up the above lines,

Modulus function

This means if the value of \(\begin{align}x\end{align}\) is greater than or equal to 0, then the modulus function takes the actual value, but if \(\begin{align}x\end{align}\) is less than 0 then the function takes minus of the actual value 'x'.


How To Calculate The Modulus Function?

The steps to calculate modulus functions are given below.

if \(\begin{align}x = -3 \end{align}\), then 

\(\begin{align}y = f(x) = f(-3) = - (-3) = 3\end{align}\), here \(\begin{align}x\end{align}\) is less than 0

if \(\begin{align}x = 4 \end{align}\), then 

\(\begin{align}y = f(x) = f(4) = 4\end{align}\), here \(\begin{align}x\end{align}\) is greater than 0

if \(\begin{align}x = 0 \end{align}\), then 

\(\begin{align}y = f(x) = f(0) = 0\end{align}\), here \(\begin{align}x\end{align}\) is equal to 0

To sum up, the modulus of a negative number and a positive number is the same number.


Graph Of Modulus Function

Now let us see how to plot the graph for a modulus function and find its domain and range.

Let us consider x to be a variable, taking values from -5 to 5

x -5 -4 -3 -2 -1 0 1 2 3 4 5
y = f(x) 5 4 3 2 1 0 1 2 3 4 5

Calculating modulus for the positive values of 'x', the line plotted in the graph is 'y = x' 

and for the negative values of 'x', the line plotted in the graph is 'y = -x'.

Modulus graph

Please note that, we can apply modulus to any real number. The range of the modulus function is the set of non-negative integer variables which is denoted as \(\begin{align}(0,\infty)\end{align}\) and the domain of modulus function is R (where R refers to the set of all positive real numbers)

As we discussed modulus is a non-negative value and by this interpretation, we can also say that modulus is the square root of square of a variable.

\(\begin{align}|x| = \sqrt{x^2}\end{align}\)

There are a few other non-negative expressions which are enumerated below.

Modulus of an expression or variable = \(\begin{align}|x|\end{align}\)

Even root of a expression or variable = \(\begin{align}x^{\dfrac{1}{2n}}\end{align}\) where \(\begin{align}n \in Z\end{align}\)

Even power of a expression or variable = \(\begin{align}x^{2n}\end{align}\) where \(\begin{align}n \in Z\end{align}\)

\(\begin{align}y = 1 - sin\:x; y = 1 - cos\:x \:as \:sin \:x ≤1\: and \:cos\:x ≤1\end{align}\)


Signum Function 

Signum function is defined as a mathematical function that gives the sign of a real number. The signum function is expressed as follows.

Signum function

The graph of the signum function is as follows.

Signum function graph


Modulus of a Complex Number

A complex number is a number which is of the form \(\begin{align}a + bi \end{align}\), where 'a' and 'b' are real numbers and 'i' is an imaginary unit. 

Modulus of a complex number \(\begin{align}z = a + bi \end{align}\)  can be defined as \(\begin{align}|z| = \sqrt{a^2 + b^2}\end{align}\)


Important Properties of a Modulus Function

Property 1:

Modulus and Equality

The modulus function always evaluates to a non-negative number for all real values of 'x'. Also equating the modulus function to a negative number is not correct.

\(\begin{align}|f(x)|=a; \:a>0⇒f(x)=±a \\|f(x)|=a;\:a=0⇒f(x)=0\\|f(x)|=a;a<0 \end{align}\)

Property 2:

Modulus and Inequality

Case 1: (If a > 0)

Inequality of a negative number

\(\begin{align} |f(x)| < a ; a> 0 \Rightarrow -a < f(x) < a \end{align}\)

Inequality for a positive number 

\(\begin{align} |f(x)| > a ; a> 0 \Rightarrow -a < f(x) > a \end{align}\)

Case 2: (If a < 0)

\(\begin{align} |f(x)| < a ; a < 0 \Rightarrow \end{align}\) - there is no solution for this.

\(\begin{align} |f(x)| > a ; a < 0 \Rightarrow \end{align}\) - this is valid for all real values of f(x).

Property 3:

If x,y are real variables, then 

\(\begin{align} |-x| = |x|  \end{align}\)

\(\begin{align} |x−y|=0⇔x=y  \end{align}\)

\(\begin{align}|x+y|≤|x|+|y| \end{align}\)

\(\begin{align}|x−y|≥||x|−|y|| \end{align}\)

\(\begin{align}|xy|=|x| \times |y| \end{align}\)

\(\begin{align}|\dfrac{x}{y}| = \dfrac{|x|}{|y|}; |y| \neq 0\end{align}\)

Now let us look into some solved modulus function questions to understand better.

 
important notes to remember
Important Notes
  1. The modulus function is also called the absolute value function and it represents the absolute value of a number. It is denoted by |x|.
  2. The domain of modulus functions is the set of all real numbers.
  3. The range of modulus functions is the set of all real numbers greater than or equal to 0.
  4. The vertex of the modulus graph y = |x|  is (0,0).

Solved Examples

Example 1

 

 

Find the modulus of x for 

a) x = -4
b) x = 6

Solution

a) x = -4

\(\begin{align}|x|  = |-4|  = - (-4) = 4 \end{align}\) 

b) x = 6

\(\begin{align}|x|  = |6| = 6 \end{align}\)

For x = -4,\(\begin{align}|-4|  = 4\end{align}\)

and

for x = 6 \(\begin{align}|6|  = 6\end{align}\)

Example 2

 

 

Solve \(\begin{align}|x+3| = 8 \end{align}\)

Solution

We will form two equations as follows.

Case 1:

Value of the modulus funciton is negative.

\(\begin{align}|x+3| = 8 \end{align}\)

\(\begin{align}-|x+3| = 8 \end{align}\)

\(\begin{align}x+3 = -8 \end{align}\)

\(\begin{align}x = -8 - 3 \end{align}\)

\(\begin{align}x = -11 \end{align}\)

Case 2 : 

Value of the modulus funciton is positive.

\(\begin{align}|x+3| = 8 \end{align}\)

\(\begin{align}x+3 = 8 \end{align}\)

\(\begin{align}x = 8 - 3 \end{align}\)

\(\begin{align}x = 5 \end{align}\)

Therefore the possible values for x in the modulus function are,

\(\begin{align}x = 5,-11\end{align}\)

x can have values of \(\begin{align}x = 5,-11\end{align}\)
Example 3

 

 

Draw the graph for \(\begin{align}y = |x +2|\end{align}\)

Solution

As per the definition of modulus function we have,

Modulus function

\(\begin{align}y = |x + 2| = x + 2, if\: x \geq 1 \\-2 - x , if \:x < 1  \end{align}\)

Let us chart a table with positive and negative values of 'x'.

y = |x+2|
-7 |-7+2| = |-5| = 5
-6 |-6+2| = |-4| = 4
-5 |-5+2| = |-3| = 3
-4 |-4+2| = |-2| = 2
-3 |-3+2| = |-1| = 1
-2 |-2+2| = |0| = 0
-1 |-1+2| = |1| = 1
0 |0+2| = |2| = 2
1 |1+2| = |3| = 3
2 |2+2| = |4| = 4
3 |3+2| = |5| = 5
4 |4+2| = |6| = 6

Plotting the graph with various values of \(\begin{align}x\end{align}\) and \(\begin{align}-x\end{align}\) we get the graph for the modulus funciton as shown below,

Modulus graph of x+2

This is the graph for modulus function x+2
Example 4

 

 

Solve \(\begin{align}|2x - 4| = 5 - x\end{align}\)

Solution

As per the definition of modulus function we have 

Modulus function

There can be two possibilities as per the modulus function.

Case 1:

\(\begin{align}-|2x - 4| = 5 - x \end{align}\)

\(\begin{align}2x - 4 = -(5 - x)\end{align}\)

\(\begin{align}2x - 4 =  -5 + x\end{align}\)

\(\begin{align}2x - x &= -5 + 4 \\ x &= -1 \end{align}\)

Case 2:

\(\begin{align}|2x - 4| = 5 - x  \\ 2x - 4 = 5 - x \\ 2x + x = 5 + 4 \\ 3x = 9 \\ x = 3\end{align}\)

\(\begin{align}x = -1\: and\: x =3\end{align}\)
 
tips and tricks
Tips and Tricks
  1. The modulus of a non-negative number and a negative number is positive. |-5| is 5 and |5| is also 5.
  2. To solve modulus equations like |x-2| = 5, make two equations like x-2 = -5 & and x - 2 = 5 to find the solution.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

 


Let's Summarize

The lesson targeted the fascinating concept of modulus function its domain and range. Hope you enjoyed learning them. Going through the solved examples and solving the ineractive questions will make you get more knowledge on the subject. You can also try out the mod function calculator to check the modulus of a number.

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FAQs on Modulus Function

1. What is modulus equation?

An equation that gives the modulus or magnitude of a given number is called modulus equation. It is denoted as y = |x|.

2. What does modulus mean?

Modulus means finding the magnitude of a positive or a negative number.

3. How do you solve modulus problems?

Applying modulus to a non-negative number and a negative number always results in the same number.

4. How do you draw a modulus function?

By taking negative values like (-1,-2,-3 ) and positive values like (1,2,3) as per the given modulus equation we can draw the modulus function.

5. Why do we use Mod?

Modulus function is used to find the magnitude of a positive or a negative number.

6. Is modulus always positive?

The modulus of a positive number is positive. The modulus of a negative number is obtained by ignoring the minus sign. Thus, modulus is always positive.

7. What is the derivative of modulus funciton?

The derivative of modulus function is x/|x|.

8. What is the range of modulus function?

The range of modulus function is the set of all non-negative numbers or simply (0, infinity).

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