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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”.

Example

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.

Example

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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
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
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 ?
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
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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