<< Chapter < Page Chapter >> Page >

Onto function (surjection)
A function f : A B is an onto function or surjection, if every element of co-domain set is the image of some element in the domain set “A”.

Onto function (surjection)

Every element of co-domain set is the image of some element in the domain set “A”.

One of the implications of surjection is that all elements of co-domain is related. It reduces the co-domain to range set. In other words, co-domain is also the range of the function.

Co-domain of "f" = Range of "f"

This equality of sets form one of the condition for testing a function to be surjection. Alternatively, we can check surjectivity by evaluating the rule of the function for the argument “x”. If the expression of “x” is valid for elements in co-domain, then the function is a surjection.

Example

Problem 2 : Consider a function defined as :

f : R R b y f x = x 3 + 1 for all x R

Determine whether the function is a surjection?

Solution : We solve the rule for argument “x” as :

y = x 3 + 1

x = y 1 1 / 3

We see that expression on the right hand side is a valid real expression for all values of “y” in "R" i.e co-domain. Hence, given function is an onto function or surjection.

Into function

We have discussed in the beigining of this module that there is possibility that some of the elements of co-domains are not related. In that case, function is said to be into function.

Onto function (surjection)
A function f : A B is an into function, if there exists element in co-domain set, which has no pre-image in the domain set “A”.

One of the implications is that all elements of co-domain are not related to elements in domain set. In other words, range of the function is subset of the co-domain :

Into function

The range of the function is subset of the co-domain .

Range of "f" Co-domain of "f"

We can check whether a given function is an into function or not by checking whether the function is an onto set or not. If the function is not an onto function, then it an into function. Alternatively, we can check the equality of co-domain and range set. If they are not equal, then the function is into function.

Into function

Problem 3 : Consider a function defined as :

f : R R b y f x = x 2 + 1 f o r a l l x R

Determine whether the function is an into function?

Solution : The rule of the function is :

y = x 2 + 1

The square of a real number is a positive number for all real number. Hence,

y = x 2 + 1 > 0

It means that images are only the right half of the real number i.e. from zero to infinity. But, the co-domain of the function is “R”. It means that left half of the co-domain i.e. from negative infinity to less than zero has no image in “A”. Therefore, the given function is an into function.

One – one onto function (bijection)

The bijection presents the most stringent condition for a function. Every element of both domain and co-domain is related to distinct element. This requirement is fulfilled when a function is both an injection and surjection.

One – one onto function (bijection)

Every element of both domain and co-domain is related to distinct element.

The injection means that every element of domain is related to a distinct element in co-domain. On the other hand, surjection means that every element of co-domain is related, both distinctly and commonly. When conditions of injection and surjection are taken together, then it is also ensured that elements of co-domains are also related to distinct elements only.

One – one onto function (bijection)

Problem 4 : Consider a function defined as :

f : A B b y f x = x - 2 x - 3

Determine domain (A) and co-domain(B) of the function so that it is a bijection.

Solution : For determining domain of the function, we inspect the given rule,

f x = x 2 x 3

We observe that the given rational function is defined for all values of “x” except for x = 3. Hence, its domain is :

Domain = R { 3 }

In order that the given function is a bijection, it should be both an injection and a surjection. For infectivity, we put :

f x = f y

x - 2 x - 3 = y - 2 y - 3

x 2 y 3 = x 3 y 2

x y 3 x 2 y + 6 = x y 2 x 3 y + 6

3 x 2 y = 2 x 3 y

x = y

Hence, function is an injection for the domain as determined above. Now, for surjection we solve the rule of the function for the argument (x),

y = x 2 x 3

x y 3 y = x 2

x y 1 = 3 y 2

x = 3 y 2 y - 1

This equation is valid for all real values of “y” except “1”. Hence, function is surjection for all real values of “y” except for “1”. Hence, co-domain for the function to be a surjection is :

Co-domain = R { 1 }

Thus, we conclude that the given function is bijection for the domain and co-domain as determined above.

Domain = R { 3 }

Co-domain = R { 1 }

Questions & Answers

what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
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

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask