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x 2 + 1 = y 2 + 1

x 2 = y 2

x = ± y

This is not an unique solution. Here, “x” is not uniquely equal to “y”. We conclude that given function is not an injection. As a matter of fact, we can infer a check on our conclusion as,

f 1 = f - 1 = 2

Thus, we see that two pre-images relate to one image, which is contradictory to the requirement of an injection.

Increasing and decreasing functions

The fact that function value is different for different arguments has an important bearing on the nature of injection plot. Consider two plots shown in the figure. In the plot shown on the left, a straight line parallel to x-axis intersects the plot only once. In the second plot, a line parallel to x-axis intersects the plot at two points for x = x 1 and x = x 2 . The function represented by second plot is not an injection as two values of arguments map to a single value of function – not two different values as required for an injection function.

One-one function (injection)

Injection graph is either increasing or decreasing.

It means that intersection plot intersects a line parallel to x-axis only once. This is possible only if the function is either (i) continuously increasing or (ii) continuously decreasing. Note the use of word “continuously”. An increasing plot, for example, if drops, then we can always find a line parallel ot x-axis, which intersects it at two points.

Hence, an injection graph is either an increasing or decreasing type. We can associate these characteristics with differential calculus. We can say that :

Either

d y d x > 0 for all x

or,

d y d x < 0 for all x

As a matter of fact the derivative can be equal to zero for certain values of "x" - not for an interval of "x". Thus, we can write the condition of increasing function : iif function is continuous and

d y d x 0 for all x ; equality holding for certain values of x

Similarly, we can write the condition of decreasing function : iif function is continuous and

d y d x 0 for all x ; equality holding for certain values of x

Many – one function

More than one pre-images of a function are related to same image.

Many - one function
A function f : A B is an many – one function, if two or more elements of domain set “A” have the same images in co-domain set “B”.

The test of condition for many-one function is easy : if a function is not one-one, then it is many-one. Alternatively, we can check literally going by the definition – whether there exist such many-one relation. A map diagram showing the relation will reveal such instances of many-one relation.

Many-one function

More than one pre-images of a function are related to same image.

Modulus function is one such many-one function. The function yields same value for positive and negative arguments of same magnitude.

f x = | x |

f - 1 = | - 1 | = 1

f 1 = | 1 | = 1

We should understand that a reverse function of the type “one to many” is barred from the very definition of function. The element of domain can be related to exactly one element in co-domain.

Onto function (surjection)

The definition of function puts the restriction on domain that every element in it is related. If we extend this restriction to co-domain also, then we get a function called “onto” or “surjection”.

Questions & Answers

what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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