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Examples on Modulus Function

What Is the Modulus Function?

The modulus of a given number describes the magnitude of the number. Modulus Function is defined as the real-valued function, say f: R -> R, where y = |x| for each x ∈ R OR f(x) = |x|. This function can be defined using the modulus operation as follows:

\(\begin{array}{l}f(x) =\left\{\begin{matrix} -x &x <0 \\ x & x \geq 0 \end{matrix}\right.\end{array} \)

For each non-negative value of x, f(x) is equal to the positive value of x. But for negative values of x, f(x) is equal to the negative value of x. Read more.

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Properties of Modulus Function

The modulus function has the following properties:

1. For any real number x, we have

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

2. ||x||=|x|

3. If a and b are positive real numbers, then

\(\begin{array}{l}a]\ x^2 \leq a^2 \Leftrightarrow |x| \leq a \Leftrightarrow -a \leq x \leq a\\ b] \ x^2 \geq a^2 \Leftrightarrow |x| \geq a \Leftrightarrow x \leq -a \\ c] \ x^2 > a^2 \Leftrightarrow |x| > a \Leftrightarrow x < -a \\ d]\ a^2 \leq x^2 \leq b^2 \Leftrightarrow a \leq |x| \leq b \Leftrightarrow x \epsilon [−b, −a] \cup [a, b].\\\end{array} \)

4.  If a is negative, then
a. |x| a, x in R

b. |x| ≤ a, x = ϕ

5.  For real numbers x and y

\(\begin{array}{l}a. |xy| = |x| |y|\\ b. |xy| = |x| |y|, y \neq 0|\\\end{array} \)
\(\begin{array}{l}c. |x+y| \leq |x| + |y| \text \ and \ |x−y| \leq |x| – |y| \text \ for \ x, \ y \geq 0 \text \ or \ x, \ y < 0, |x+y| = |x| + |y|\\\end{array} \)
\(\begin{array}{l}d. |x+y| \geq |x| − |y| |x+y| \text \ and |x−y| \geq |x| − |y| \text \ for \ x, \ y \geq 0 \text \ and \ |x| \geq |y| y \text \ or \ x, \ y < 0 \text \ and \ |x| \geq |y|, |x−y| = |x| − |y|\\\end{array} \)
.

6. The absolute value has the following four fundamental properties (a and b are real numbers):

\(\begin{array}{l}1] \ {\displaystyle |a|\geq 0} \text \ Non-negativity\\ 2] \ {\displaystyle |a|=0\iff a=0} \text  \ Positive-definiteness\\ 3] \ {\displaystyle |ab|=|a|\,|b|} \text \ Multiplicativity\\ 4] \ {\displaystyle |a+b|\leq |a|+|b|} \text \ Subadditivity, \text specifically \ the \ triangle \ inequality\\\end{array} \)

7. Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.

a] ||a||=|a|: Idempotence (The absolute value of the absolute value is the absolute value)

\(\begin{array}{l} b] \ {\displaystyle |-a|=|a|} \text \ Evenness \ (reflection \ symmetry \ of \ the \ graph)\\ c] \ {\displaystyle |a-b|=0\iff a=b} \text \ Identity \ of \ indiscernibles \ (equivalent \ to \ positive-definiteness) \\ d] \ {\displaystyle |a-b|\leq |a-c|+|c-b|} \text \ Triangle \ inequality \ (equivalent \ to \ subadditivity) \\ e] \ {\displaystyle \left|{\frac {a}{b}}\right|={\frac {|a|}{|b|}}\ }\ (\text\ if \ {\displaystyle b\neq 0}) \text \ Preservation \ of \ division \ (equivalent \ to \ multiplicativity) \\ f] \ {\displaystyle |a-b|\geq {\big |}\,|a|-|b|\,{\big |}} \text\ Reverse \ triangle \ inequality \ (equivalent \ to \ subadditivity)\\\end{array} \)

8. Two other useful properties concerning inequalities are as follows:

\(\begin{array}{l}1] \ {\displaystyle |a|\leq b\iff -b\leq a\leq b}\\ 2] \ {\displaystyle |a|\geq b\iff a\leq -b\ } \text \ or \ {\displaystyle a\geq b}\\\end{array} \)

Graph of Modulus Function

Modulus Graph Function

Video Lesson

Solved Examples on Modulus

Example 1: Solve modulus and find the interval of x for |x2 – 5x + 6|

Solution: |x2 – 5x + 6| = |(x – 2)(x – 3)| = |f(x)|

As per the modulus definition,

|f(x)| = f(x); if f(x) is positive

| f(x) |= -f(x); if f(x) is negative

f(x) = (x – 2)(x – 3) is positive or zero when x = (- ∞, 2] ∪ [3, ∞)

f(x) = (x – 2)(x – 3) is negative when x = (2, 3)

So, |x2 – 5x + 6| = (x2 – 5x + 6) when x = (-∞, 2] ∪ [3, ∞) and

|x2 – 5x + 6| = -(x2 – 5x + 6) when x = (2, 3)

Example 2: If |x2 – 5x + 6| + |x2 – 8x + 12| = 0. Find x.

Solution:

Every modulus is a non-negative number, and if two non-negative numbers add up to get zero, then individual numbers itself equal to zero simultaneously.

x2 – 5x + 6 = 0 for x = 2 or 3

x2 – 8x + 12 = 0 for x = 2 or 6

Both the equations are zero at x = 2

So, x = 2 is the only solution for this equation.

Example 3: Solve for x, |x – 1| – |x – 2| = 10

Solution: Here, the critical points are 1 and 2.

Let us check for the values less than 1, between 1 and 2, and greater than 2.

Case 1: For x ≤ 1

-(x – 1) – {-(x – 2)} = 10

or -x + 1 + x – 2 = 10

or -1 = 10 (which is not possible)

Case 2: x ∈ (1, 2)

(x – 1) – {-(x – 2)} = 10

or x – 1 + x – 2 = 10

or 2x – 3 = 10

or x = 13/2

In this case, x ∈ (1, 2), so 6.5 is not a solution.

Case 3: x ≻ 2

(x – 1) – (x – 2) = 10

=> 1 = 10 (which is not possible)

So, this equation has no solution.

Example 4: Draw the graph of |sin(x) + cos(x)| in the interval x ϵ [0, π].

Solution:

|sin(x) + cos(x)| = |√2 sin(x + π/4)|

First, make the graph of the function, then take the modulus.

Here, at the place of sinx, we have √2 sin(x + π/4), so the graph will be of type sin x only, but the graph will be starting from x = -π/4.

y = sin(x) + cos(x)

Modulus JEE

Take all negative y values to the positive y side, and positive y remains the same.

y = |sin(x) + cos(x)| = |√2 sin(x + π/4)|

Example on Modulus

Example 5: Find the domain and range of the below function.
a. y = |1−x|
b. y = 2 − |1 − x|
c. y = 2√x − |x|

Solution:

a. y = |1 − x|
Clearly, this is defined for x ∈ R. So, Domain is R.
Now, |1 − x| ≥ 0 for all x ∈ R
So, the range is [0, ∞).

b. y = 2 − |1 − x|
Clearly, this is defined for x ∈ R.

So, the domain is R.
Now, |1 − x| ≥ 0 for all x ∈ R
or
−|1 − x| ≤ 0
2 − |1 − x| ≤ 2 for all x∈R
So, the range is (−∞, 2].

c. y = 2√x − |x|
Now, x− |x| = {x − x = 0 if x ≥ o, x + x = 2x if x ≤ 0

Therefore, 2√x − |x| is undefined for all x ∈ R.
So, the domain of the function is ϕ.

Frequently Asked Questions

Q1

What do you mean by a modulus function?

A modulus function is a function which gives the absolute value of a number or variable. The outcome of the modulus function is always positive.

Q2

How do you find the modulus of a negative number?

The modulus of a negative number is the number without the minus sign. For example, |-4| = 4.

Q3

Is the modulus of a number always positive?

Yes, the modulus of any number is always positive.

Q4

What is the range of the modulus function?

The range of the modulus function is the set of non-negative real numbers, i.e., (0,∞).

Q5

What is the shape of the graph of modulus function f(x) = |x|?

The graph of modulus function f(x) = |x| is a V-shaped graph.

Test your Knowledge on examples on modulus function

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