# 2.2 Functions  (Page 2/3)

 Page 2 / 3

$\text{Domain of "f"}=A$

$\text{Co-domain of "f"}=B$

$\text{Range of "f"}=\text{Set of images}=\left\{f\left(x\right):x\in A\right\}$

## Equal functions

Two functions are equal, if each ordered pair in one of the two functions is uniquely present in other function. It means that if “g” and “h” be two equal functions, then :

$g\left(x\right)=h\left(x\right)\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{1em}{0ex}}“x”$

Two functions g(x) and h(x) are equal or identical, if all images of two functions are equal. Further, we can visualize equality of two functions in a negative context. If there exists “x” such that g(x)≠ h(x), then two functions are not equal. We state this symbolically as :

$\text{If}\phantom{\rule{1em}{0ex}}g\left(x\right)\ne h\left(x\right),\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}f\ne h$

The important question, however, is that whether equality of functions in terms of equality of images is a sufficient condition? We can see here that two functions can meet the stated condition even if they are constituted by different sets of ordered pair. There may be additional ordered pairs, which are present in one, but not in other.

In order to remove such possibilities, two equal functions should have same domain. This will ensure that set of ordered pairs in two functions are same. We conclude this discussion by saying that two functions are equal, iff

• g(x) = h(x) for all “x”
• Domain of “f” = Domain of “h”

It is clear that equality of functions, however, do not require that co-domains be equal.

## Real function

If the range of a function is a set of real numbers, then the function is called “real valued function”. In other words, if the range of a function is either the set “R” or its subset, then it is a real valued function. We should emphasize here that “R” denotes set of real number and it is not the symbol for relation, which is also denoted as “R”.

Further, we distinguish “real valued function” from “real function”. The very terminology is indicative of the difference. The term “real valued function” means that the value of function i.e. image is real. It does not say anything about “pre-image”. Now, there can be a function, which accepts non-real complex numbers, but maps to a real value.

On the other hand, a real function has both image and pre-image as real numbers. It follows then that the domain of a “real function” is also either a set or subset of real numbers.

Real function
A function is a real function, if its domain and range are either “R” or subset of “R”.

Our discussion from this point onwards in the course relates to real function only – unless otherwise stated.

## Interpretation of function relation

It is intuitive to find similarity of an algebraic equation to the “rule” of a function. Consider an equation,

$y={x}^{2}+1$

This equation is valid for all real values of “x”. The set of real values of “x”, belongs to set “R”. The set of values of “y” also belongs to set of “R”. On the other hand, the equation itself is the rule that maps two sets comprising of values of “x” and “y”.

Alternatively, we can write the rule also as :

$⇒f\left(x\right)={x}^{2}+1$

In terms of rule, we define function, saying that :

$f:R\to R\phantom{\rule{1em}{0ex}}by\phantom{\rule{1em}{0ex}}f\left(x\right)={x}^{2}+1$

We read it as : “f” is a function from “R” to “R” by the rule given by $f\left(x\right)={x}^{2}+1$ .

From this description, we think a function as a relation, which is governed by a specified rule. The rule relates two sets known as domain and co-domain, which are sets of real numbers. One of the quantities “x” is independent of other quantity “y”. The other quantity “y” is a dependent on quantity “x”. In plain words, one of the interpretations is that function relates dependent and independent variables. As a matter of fact, we would attach additional meanings to the concept of function as we proceed to study it in details.

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
Abigail
Do somebody tell me a best nano engineering book for beginners?
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
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's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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