Mathematics for Biotechnology and Data Science
Chapter 3: Application of Set Theory
Chapter 6: MAPPING OR FUNCTIONS
Chapter 8: Inverse of a Function
8.1. Inverse Function
Let f be a one-one
function from A onto B. Since f is onto, therefore, 
, there exists 
such that 
and since f is one-one, therefore this element
x is unique. Thus, a function 
can be defined from B onto A such that 
. This function is called the inverse function of 
and is denoted by 
.
Thus 
such that 
iff .
Example 1: 
How to find the inverse of a given function.
To find the inverse of
the function 
, express x in terms of
y. then value of x in terms of y will be 
. Now put x in place of
y in to get 
.
Invertible function: A function f is said to be invertible if its
inverse exists i.e., if exists.
8.2. Properties of Inverse of a function.
i. If
is an invertible
function, then is one- one onto.
ii. Inverse function, if it exists, is also a one-one onto function.
iii. Inverse function, if it exists, is unique.
iv.
If 
is a bijection and 
is the inverse of 
then 
and 
, where 
and 
are the identity functions on the sets A and B
respectively.
v. If
and be two functions such that and , then and are bijections and 
.
vi.
If and 
are one-one onto functions then 
.
Example
1: let 
and 
. 
is a function defined
as 
. Write down 
as a set of ordered pairs.
Solution: we have given
Here
distinct elements of A have distinct images in B, therefore, is one-one.
Alos
each element of B is the image of some element of A, therefore is onto. Thus is one-one onto. 
exists.
Now
Example
2: Show that 
given by 
is invertible and it is inverse of itself.
Solution: To show 
is invertible, it is sufficient to show that
it is one-one onto.
is one-one: Let 
such that 
.
Now,
Thus, 
for all 
.
Therefore, is one-one.
is onto: Let y be an arbitrary element
of co-domain 
.
Now, 
Thus, for each 
, there exist 
Such that . Therefore, is onto.
Hence is one-one onto and therefore, it is invertible.
To find :
Let 
Clearly, 
, for 
Hence, is inverse of itself.
Example 3: Show that 
given by 
, is one-one. Find the
inverse of the function 
Solution:
Given 
defined by 
is one-one:
Let 
such that 
Now,
Hence, 
is one-one.
is onto: Let range of 
Since f is one-one,
therefore, inverse of the function exist.
To find 
: Let
Example 4: Consider 
given by 
, where 
is the set of all non-negative real numbers,
Show the f is invertible with 
.
Solution:
Given given by ,
To test
whether is one-one
Let 
such that 
Now, 
Hence 
is
one-one.
To test
whether is onto.
Let y be any arbitrary
element of 
Let
Now, 
Hence is onto.
To find 
Let 
Continue .........
