Let A = {2, 3, 4}, B = {a, b, c} and the function f : A → B is
defined by f = {(2, a), (3, b), (4, c)}.

Here,

2 ∈ A, f(2) = a ∈ B

3 ∈ A, f(3) = b ∈ B

4 ∈ A, f(4) = c ∈ B

Also domain of f = {2, 3, 4} and range of f = {a, b, c}.

∴ f is one-to-one onto function.

Now we define a relation g : B → A so that g = {(a, 2), (b, 3),
(c, 4)}.

Here,

a ∈ B, g(a) = 2

b ∈ B, g(b) = 3

c ∈ B, g(c) = 4

So, g is also one-to-one onto function.

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The function g is defined from B to A is said to be the **inverse**
of f is denoted by f^{-1}. But for every function, there may not exist
its inverse. Let us study the following examples.

__1. f = {(2, 3), (3, 3), (4, 5)}__.

Here, f is “many to one function.”

Here a relation from B to A is not a function as 3 ∈ B corresponds with more than one element of A. So, f^{-1}
does not exist.

__2. f = {(2, 3), (4, 8), (5, 2)}__.

Here, f is a one-to-one into function.

Here, a relation from B to A is not a function because 9 ∈ B does not correspond with any element of A. So, f^{-1}
does not exist.

Hence, from the above examples, it is clear that to define an inverse
function f must be a one-to-one onto function.

**Definition of Inverse
Function:**

Let f : A → B be a one-to-one onto function. Then a function
from B to A which associates each element of B with a unique element of A is
called the **inverse function** of f. The inverse function from B to A is denoted ^{-1} : B → A.

**Worked Out Examples:**

**Example 1:** Let A = {a, b, c} and f : A → A is defined as f = {(a, c), (b, a),
(c, b)}, find f^{-1}.

**Solution: **

Here,

f = {(a, c), (b, a), (c, b)}

Clearly, the function f is one-to-one onto, hence f^{-1}
exists and

f^{-1} = {(c, a), (a, b), (b, c)}.

**Example 2:** Let f : R → R be defined by f(x) = 2x – 3, find the formula that
defines f^{-1}.

**Solution: **

Here,

f : R → R and x is a variable, f is a one-to-one function, f^{-1}
exists.

y = f(x) = 2x – 3,

To find the inverse function we interchange the role of x and y,

i.e. x = 2y – 3

or, 2y = x + 3

or, y = (x + 3)/2

Hence, inverse function y = f^{-1}(x) = (x + 3)/2

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