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We must also understand that this new function, “ f - 1 ”, gives the perspective of relation from co-domain to domain of the given function “f”. However, new function “ f - 1 ” is read from its new domain to its new co-domain. After all this is how a function is read. This simply means that domain and co-domain of the function “f” is exchanged for “ f - 1 ”.

Inverse function

The domain and co-domain sets are exchanged.

Further, inverse function is inverse of a given function. Again by definition, every element of domain set of the given function “f” is also related to exactly one element of in its co-domain. Thus, there is bidirectional requirement that elements of one set are related to exactly one element of other set. Clearly, this requirement needs to be fulfilled, before we can define inverse function.

In other words, we can define inverse function, " f - 1 ", only if the given function is an injection and surjection function (map or relation) at the same time. Hence, iff function, “f” is a bijection, then inverse function is defined as :

f - 1 : A B by f - 1 x for all x A

We should again emphasize here that sets “A” and “B” are the domain and co-domain respectively of the inverse function. These sets have exchanged their place with respect to function “f”. This aspect can be easily understood with an illustration. Let a function “f” , which is a bijection, be defined as :

Let A = {1,2,3,4} and B={3,6,9,12}

f : A B by f x = 3 x for all x A

Then, the function set in the roaster form is :

f = { 1,3 , 2,6 , 3,9 , 4,12 }

This function is clearly a bijection as only distinct elements of two sets are paired. Its domain and co-domains are :

Domain of “f” = { 1,2,3,4 }

Co-domain of “f” = { 3,6,9,12 }

Now, the inverse function is given by :

f - 1 : A B b y f x = x 3 for all x A

where A = { 3,6,9,12 }

In the roaster form, the inverse function is :

f - 1 = { 3,1 , 6,2 , 9,3 , 12,4 }

Note that we can find inverse relation by merely exchanging positions of elements in the ordered pairs. The domain and co-domain of new function “ f - 1 ” are :

Domain of f - 1 = { 3,6,9,12 }

Co−domain of f - 1 = { 1,2,3,4 }

Thus, we see that the domain of inverse function “ f - 1 ” is co-domain of the function “f” and co-domain of inverse function “ f - 1 “ is domain of the function “f”.


Problem 1: A function is given as :

f : R R by f x = 2 x + 5 for all x R

Construct the inverse rule. Determine f(x) for first 5 natural numbers. Check validity of inverse rule with the values of images so obtained. Find inverse function, if it exists.

Solution : Following the illustration given earlier, we derive inverse rule as :

y = 2 x + 5

x = y - 5 2

Changing notations,

f - 1 x = x 5 2

The images i.e. corresponding f(x), for first five natural numbers are :

f 1 = 2 x + 5 = 7 ; f 2 = 9 ; f 3 = 11 ; f 4 = 13 a n d f 5 = 15

Now, the corresponding pre-images, using inverse rule for two values of images are :

f - 1 7 = 7 - 5 2 = 1

f - 1 11 = 11 - 5 2 = 3

Thus, we see that the inverse rule correctly determines the pre-images as intended. Now, in order to find inverse function, we need to determine that the given function is an injection and surjection. For injection, let us assume that “ x 1 ” and “ x 2 ” be two different elements such that :

f x 1 = f x 2

2 x 1 + 5 = 2 x 2 + 5

x 1 = x 2

This means that given function is an injection. Now, to prove surjection, we solve the rule for “x” as :

x = y - 5 2

We see that this equation is valid for all values of “R” i.e. all values in the co-domain of the given function. This means that every element of the co-domain is related. Hence, given function is surjection. The inverse function, therefore, is given as :

f - 1 : R R b y f x = x 5 2 for all x R

Properties of inverse function

There are few characteristics of inverse function that results from the fact that it is inverse of a bijection. We can check the validity of these properties in terms of the example given earlier. Let us define a bijection function as defined earlier :

Let A = { 1,2,3,4 } and B = { 3,6,9,12 }

f : A B by f x = 3 x for all x A

Inverse function is unique function

This means that there is only one inverse function. For the given function the inverse function is :

f - 1 : A B b y f x = x 3 f o r a l l x A

where A = { 3,6,9,12 }

In the roaster form, the inverse function is :

f - 1 = { 3,1 , 6,2 , 9,3 , 12,4 }

This inverse function is unique to a given bijection.

Inverse function is bijection

We see that inverse comprises of ordered pairs such that elements of domain and co-domain are distinctly related to each other.

f - 1 = { 3,1 , 6,2 , 9,3 , 12,4 }

This mean that the inverse function is bijection.

Graph of inverse function

If a function is bijection, then the inverse of function exists. On the other hand, a function is bijection, if it is both one-one and onto function. We know that one-one function is strictly monotonic in its domain. Hence, an onto function is invertible, if its graph is strictly monotonic i.e. either increasing or decreasing.

In order to investigate the nature of the inverse graph, let us consider a plot of an invertible function, “f(x)”. Let (a,b) be a point on the plot. Then, by definition of an inverse function, the point (b,a) is a point on the plot of inverse function, if plotted on the same coordinate system.

y = f ( x )

y = f - 1 ( x )

By geometry, the line joining points (a,b) and (b,a) is bisected at right angles by the line y = x. It means that two points under consideration are object and image for the mirror defined by y=x. This relationship also restrains that two plots can intersect only at line y = x.

Graph of inverse function

The points on two plots are object and image for the mirror defined by y=x.


Problem 2 : Two functions, inverse of each other, are given as :

f x = x 2 x + 1

f - 1 x = 1 2 + x 3 4

Find the solution of the equation :

x 2 x + 1 = 1 2 + x 3 4

Solution :

Statement of the problem : The given functions are inverse to each other, which can intersect only at line defined by y = x. Clearly, the intersection point is the solution of the equation.

y = f x = x

x 2 x + 1 = x x 2 2 x + 1 = 0

x 1 2 = 0

x = 1

This is the answer. It is interesting to know that we can also proceed to find the solution by working on the inverse function. This should also give the same result as given functions are inverse to each other.

y = f - 1 x = x

1 2 + x 3 4 = x

x 3 4 = x 1 2

Squaring both sides,

x 3 4 = x 1 2 2 = x 2 + 1 4 x

x 2 + 1 4 x x + 3 4 = 0 x 2 2 x + 1 = 0

x = 1

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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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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