# 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.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x