Modulus Functions, Properties of Modulus Function

Functions are one of the basic concepts in mathematics that have numerous applications in the real world. Be it mega skyscrapers or super-fast cars, their modeling requires methodical application of functions. Almost all real-world problems are formulated, interpreted, and solved using functions. Modulus functions help in solving modulus inequalities.

In this article, we will cover the concepts of the modulus function. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

Modulus Function:

The function f: RR defined by f(x) = |x| for each x R is called the modulus function.

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

$|\mathrm{x}|, \mathrm{x} \in \mathbb{R}=\left\{\begin{array}{cc}x, & x \geq 0 \\ -x, & x<0\end{array}\right.$

Range $\in[0, \infty)$

Modulus Function Graph

Now let us see how to plot the graph for a modulus function. Let us consider $x$ to be a variable, taking values from -5 to 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 ' $y=-x^{\prime}$.

Modulus Equations: Properties
If $a>0$
1. $|x|=a$, then $x=a,-a$
2. $|x|=|-x|$
3. $|x|^2=x^2$
4. If $|\mathrm{x}|=\mathrm{x}$, then $\mathrm{x}>0$ or $\mathrm{x}=0$
5. If $|x|=-x$, then $x<0$ or $x=0$
6. $|f(x)|=|g(x)|$, then $f(x)=g(x)$ or $f(x)=-g(x)$

Modulus inequalities
These deal with the inequalities ( $<,>, \leq, \geq$ ) on expressions with absolute value sign.
Properties
If $a, b>0$, then

Summary

we concluded that the modulus function is a versatile and widely used function in mathematics that provides a measure of magnitude irrespective of sign. Its properties make it essential for various mathematical operations and applications across different fields, including geometry, complex analysis, and optimization. Understanding and utilizing the modulus function is crucial for solving problems that involve distance, error, and absolute differences.

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Solved Examples Based On the Modulus Functions:

Example 1: If $f(x)=3|x|$ . Then the range of f(x) is

1) [0,)

1.$\begin{aligned}
& |x| \leq \mathrm{a} \Rightarrow \mathrm{x}^2 \leq \mathrm{a}^2 \\
& \Rightarrow-\mathrm{a} \leq \mathrm{x} \leq \mathrm{a}
\end{aligned}$

2.$\begin{aligned}
& |x| \geq a \Rightarrow x^2 \geq a^2 \\
& \Rightarrow x \leq-a \text { or } x \geq a
\end{aligned}$

3.$\begin{aligned}
& \mathrm{a} \leq|\mathrm{x}| \leq \mathrm{b} \Rightarrow \mathrm{a}^2 \leq \mathrm{x}^2 \leq \mathrm{b}^2 \\
& \Rightarrow \mathrm{x} \in[-\mathrm{b},-\mathrm{a}] \cup[\mathrm{a}, \mathrm{b}]
\end{aligned}$

4. $|x+y|=|x|+|y| \Leftrightarrow x y \geq 0$.
5. $|x-y|=|x|-|y| \Rightarrow x \cdot y \geq 0$ and $|x| \geq|y|$
6. $|x \pm y| \leq|x|+|y|$
7. $|x \pm y| \geq|| x|-| y||$

2) $[3, \infty)$
3) $R$
4) $[1, \infty)$

Solution:
We can find a range of $|x|$ by simple manipulation
As $0 \leq|x|<\infty$
Multiplying all 3 sides by 3

$0 \leq 3|x|<\approx$

So, the range is the same i.e $[0, \infty)$
Hence, the answer is the option 1.
Example 2: If $f(x)=-|x|+9$. Then the range of $\mathrm{f}(\mathrm{x})$ is
1) $[8, \infty)$
2) $(9, \infty)$
3) $(-\infty, 0)$
4) $(-\infty, 9]$

Solution:

We can find the range of $|x|$ by simple manipulation

$$
\text { As } 0 \leq|x|<\infty
$$

multiply all 3 sides by (-1)

$$
-\infty<-|x| \leq 0
$$

Now, adding 9 to all 3 sides

$$
-\infty<-|x|+9 \leq 9
$$

So, the range is $(-\infty, 9]$

Hence, the answer is the option 4.

Example 3: If $f(x)=|x|+x$, then what is the range of $\mathrm{f}(\mathrm{x})$ ?
1) $[0, \infty 0)$
2) $R$
3) $\{0\}$
4) $(-\infty, 0]$

Solution:
Case 1: If $x \geq 0 ; f(x)=x+x=2 x$
In this case, $x \geq 0$
Multiplying both sides by $2,2 x \geq 0$
Case 2: If $x<0 ; f(x)=-x+x=0$
Hence range $=[0, \infty)$
Hence, the answer is the option 1.
Example 4: The number of real solutions of the equation, $x^2-|x|-12=0$ is:
Solution:
Let $|x|=t$
$\Rightarrow t^2-t-12=0$

Example 5: If $\left|x^2-9\right|+\left|x^2-4\right|=5$, then the set of values of x is:
1) $(-\infty,-3) \cup(3, \infty)$
2) $(-\infty,-2) \cup(3, \infty)$
3) $(-\infty, 3)$
4) $[-3,-2] \cup[2,3]$

Solution:

$\begin{aligned}
& \left|x^2-9\right|+\left|x^2-4\right|=5 \\
& \left|x^2-9\right|+\left|x^2-4\right|=\left|\left(x^2-9\right)-\left(x^2-4\right)\right| \\
& \{\because|a|+|b|=|a-b| \Leftrightarrow a . b \leq 0\}
\end{aligned}$

So, $\left(x^2-9\right)\left(x^2-4\right) \leq 0$

$x \in[-3,-2] \cup[2,3]$
$\Rightarrow(t-4)(t+3)=0$
$\Rightarrow t=4$ or $t=-3$
$\Rightarrow|x|=4$ or $|x|=-3$
$\Rightarrow x=4,-4$ or $x \epsilon \phi$
$\Rightarrow x=4,-4$

Hence, the answer is 2.

Hence, the answer is the option (4).

Frequently Asked Questions(FAQ)-

1. What is the modulus function?

Ans: The modulus function, denoted as $|x|$, gives the absolute value of a number $x$. It is defined as:
- $|x|=x$ if $x \geq 0$
- $|x|=-x$ if $x<0$
2: How do you graph the modulus function?
Ans: The graph of $y=|x|$ is a V-shaped graph that intersects the origin $(0,0)$. For $x \geq 0$, it is the line $y=x$, and for $x<0$, it is the line $y=-x$.

3: What is the modulus of a complex number?

Ans: For a complex number $z=a+b i$ (where $a$ and $b$ are real numbers and $i$ is the imaginary unit), the modulus is $|z|=\sqrt{a^2}+b^2$

4: What is the difference between modulus and absolute value?

Ans: The terms "modulus" and "absolute value" are often used interchangeably when referring to real numbers. However, "modulus" is more commonly used in the context of complex numbers, whereas "absolute value" is typically used for real numbers.

5: Can the modulus function be used in calculus?

Ans: Yes, the modulus function is used in calculus, especially in dealing with limits, integrals, and series. The properties of the modulus function can simplify the analysis of the behavior of functions and sequences.