# 2.3 Function types  (Page 3/3)

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

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.

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

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 :

$f:R\to R\phantom{\rule{1em}{0ex}}by\phantom{\rule{1em}{0ex}}f\left(x\right)={x}^{3}+1\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{1em}{0ex}}x\in R$

Determine whether the function is a surjection?

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

$y={x}^{3}+1$

$⇒x={\left(y-1\right)}^{1/3}$

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 $f:A\to B$ 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 :

$\text{Range of "f"}\subset \text{Co-domain of "f"}$

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 :

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

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.

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 :

$f:A\to B\phantom{\rule{1em}{0ex}}by\phantom{\rule{1em}{0ex}}f\left(x\right)=\frac{x-2}{x-3}$

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,

$f\left(x\right)=\frac{x-2}{x-3}$

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

$\mathrm{Domain}=R-\left\{3\right\}$

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

$f\left(x\right)=f\left(y\right)$

$⇒\frac{x-2}{x-3}=\frac{y-2}{y-3}$

$⇒\left(x-2\right)\left(y-3\right)=\left(x-3\right)\left(y-2\right)$

$⇒xy-3x-2y+6=xy-2x-3y+6$

$⇒-3x-2y=-2x-3y$

$⇒x=y$

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),

$⇒y=\frac{x-2}{x-3}$

$⇒xy-3y=x-2$

$⇒x\left(y-1\right)=3y-2$

$⇒x=\frac{3y-2}{y-1}$

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 :

$\mathrm{Co-domain}=R-\left\{1\right\}$

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

$\mathrm{Domain}=R-\left\{3\right\}$

$\mathrm{Co-domain}=R-\left\{1\right\}$

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
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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
ya I also want to know the raman spectra
Bhagvanji
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?
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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
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Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x

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