# 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.

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
what is variations in raman spectra for nanomaterials
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
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
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