# 3.7 Modulus function  (Page 2/2)

 Page 2 / 2

For the sake of understanding, we consider a non-negative number "2" equated to modulus of independent variable "x" like :

$|x|=2$

Then, the values of “x” satisfying this equation is :

$⇒x=±2$

It is intuitive to note that values of "x" satisfying above equation is actually the intersection of modulus function "y=[x]" and "y=2" plots as shown in the figure.

Further, it is easy to realize that equating a modulus function to a negative number is meaningless. The modulus expression "|x|" always evaluates to a non-negative number for all real values of "x". Observe in the figure above that line "y=-2" does not intersect modulus plot at all.

We express these results in general form, using an expression f(x) in place of "x" as :

$|\mathrm{f\left(x\right)}|=a;\phantom{\rule{1em}{0ex}}a>0\phantom{\rule{1em}{0ex}}⇒\mathrm{f\left(x\right)}=±a$

$|\mathrm{f\left(x\right)}|=a;\phantom{\rule{1em}{0ex}}a=0\phantom{\rule{1em}{0ex}}⇒\mathrm{f\left(x\right)}=0$

$|\mathrm{f\left(x\right)}|=a;\phantom{\rule{1em}{0ex}}a<0\phantom{\rule{1em}{0ex}}⇒\text{There is no solution of this equality}$

## Modulus as distance

The modulus of an expression “x-a” is interpreted to represent “distance” between “x” and “a” on the real number line. For example :

$|x-2|=5$

This means that the variable “x” is at a distance “5” from “2”. We see here that the values of “x” satisfying this equation is :

$⇒x-2=±5$

Either

$⇒x=2+5=7$

or,

$⇒x=2-5=-3$

The “x = 7” is indeed at a distance “5” from “2” and “x=-3” is indeed at a distance “5” from “2”. Similarly, modulus |x|= 3 represents distance on either side of origin.

## Modulus and inequality

Interpretation of inequality involving modulus depends on the nature of number being compared with modulus.

Case 1 : a>0

The values of x that satisfy "less than (<)" inequality lies between intervals defined between -a and a, excluding end points of the interval. On the other hand, values of x that satisfy "greater than (>)" inequality lies in two disjointed intervals.

$|x|0\phantom{\rule{1em}{0ex}}⇒-a

$|x|>a;\phantom{\rule{1em}{0ex}}a>0\phantom{\rule{1em}{0ex}}⇒x<-a\phantom{\rule{1em}{0ex}}\mathrm{or}\phantom{\rule{1em}{0ex}}x>a\phantom{\rule{1em}{0ex}}⇒x\in \left(\mathrm{-\infty }\mathrm{,a}\right)\cup \left(\mathrm{a,}\infty \right)$

Important aspect of these inequalities is that they can be used to express intervals in compact form. For example, range of cosecant trigonometric function is $x\in \left(\mathrm{-\infty }\mathrm{,-1}\right]\cup \left[\mathrm{1,}\infty \right\}$ . Equivalently, we can write this interval as |x|≥1.

We extend these results to an expression as :

$|\mathrm{f\left(x\right)}|0\phantom{\rule{1em}{0ex}}⇒-a<\mathrm{f\left(x\right)}

For inequality involving greater than comparison with a positive number represents union of two separate intervals.

$|\mathrm{f\left(x\right)}|>a;\phantom{\rule{1em}{0ex}}a>0\phantom{\rule{1em}{0ex}}⇒\mathrm{f\left(x\right)}<-a\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\mathrm{f\left(x\right)}>a$

Case 2 : a<0

Modulus can not be equated to negative number as modulus always evaluates to non-negative number. Clearly, modulus of an expression or variable can not be less than a negative number. However, modulus function is always greater than negative number. Hence, we conclude that :

$|x|

$|x|>a;\phantom{\rule{1em}{0ex}}a<0\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\text{This inequality is valid for all real values of x}\phantom{\rule{1em}{0ex}}$

Extending these results to expression, we have :

$|\mathrm{f\left(x\right)}|

$|\mathrm{f\left(x\right)}|>a;\phantom{\rule{1em}{0ex}}a<0\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\text{This inequality is valid for all real values of f(x)}\phantom{\rule{1em}{0ex}}$

Problem : Find the domain of the function given by :

$f\left(x\right)=\frac{x}{\sqrt{\left(|x|-x\right)}}$

Solution : The function is in rational form. The domain of the function in the numerator is "R". We are, now, required to find the value of “x” for which denominator is real and not equal to zero. Now, expression within square root is a non-negative. However, as the function is in denominator, it should not evaluate to zero either. It means that the expression within square root is positive :

$⇒|x|-x>0$

$⇒|x|>x$

If “x” is a non-negative number, then by definition, "|x| = x". This result, however, is contradictory to the inequality given above. Hence, “x” can not be non-negative. When "x" is a negative number, then the inequality holds as modulus of a real variable is always greater than negative number. It means that "x" is a negative number :

$⇒x<0$

Thus, domain of the given function is equal to intersection of "R" and interval "x<0". It is given by :

$\text{Domain}=\left(-\infty ,0\right)$

## Additional properties of modulus function

Here, we enumerate some more properties of modulus of a real variable i.e. modulus of a real number.

Let x,y and z be real variables. Then :

$|-x|=|x|$ $|x-y|=0\phantom{\rule{1em}{0ex}}⇔x=y$ $|x+y|\le |x|+|y|$ $|x-y|\ge ||x|-|y||$ $|xy|=|x|X|y|$ $|\frac{x}{y}|=\frac{|x|}{|y|};\phantom{\rule{1em}{0ex}}|y|\ne 0$

Modulus |x-y| represents distance of x from y. Also, we know that sum of two sides of a triangle is greater than third side. Combining these two facts, we write a general property for modulus involving real numbers as :

$|x-y|<|x-z|+|z-y|$

## Square function

There is striking similarity between modulus and square function. Both functions evaluate to non-negative values.

$y=|x|;\phantom{\rule{1em}{0ex}}y\ge 0$

$y={x}^{2};\phantom{\rule{1em}{0ex}}y\ge 0$

Their plots are similar. Besides, they behave almost alike to equalities and inequalities. We shall not discuss each of the cases as done for the modulus function, but with a specific number (4 or -4). We shall enumerate each of the possibilities, which can be easily understood in the background of discussion for modulus function.

1: Equality

${x}^{2}=4\phantom{\rule{1em}{0ex}}⇒x=±2$

${x}^{2}=-4\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\text{No solution}$

2: Inequality with non-negative number

A. Less than or less than equal to

${x}^{2}<4\phantom{\rule{1em}{0ex}}⇒-2

B. Greater than or greater than equal to

${x}^{2}>4\phantom{\rule{1em}{0ex}}⇒x<-2\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x>2\phantom{\rule{1em}{0ex}}⇒\left(-\infty ,-2\right)\phantom{\rule{1em}{0ex}}\cup \phantom{\rule{1em}{0ex}}\left(2,\infty \right)$

3: Inequality with negative number

A. Less than or less than equal to

${x}^{2}<-4\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\text{No solution}$

B. Greater than or greater than equal to

${x}^{2}>-4\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\text{Always true}$

## Acknowledgment

Author wishes to thank Mr. Ritesh Shah for making suggestion to remove error in the module.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x

#### Get Jobilize Job Search Mobile App in your pocket Now! By Laurence Bailen By Sandy Yamane By OpenStax By OpenStax By Anh Dao By Janet Forrester By JavaChamp Team By Tony Pizur By Jonathan Long By