Finding Inverse

Chapter 1 Class 12 Relation and Functions
Concept wise

We use two methods to find if function has inverse or not

1. If function is one-one and onto, it is invertible.
2. We find g, and check fog = I Y and gof = I X

We discussed how to check one-one and onto previously.

Let’s discuss the second method

## We find g, and check fog = I Y and gof = I X

Steps are

Checking inverse of f : X → Y

Step 1 : Calculate g: Y → X

Step 2 : Prove gof = I X

Step 3 : Prove fog = I Y

## Example

Let f : N → Y,

f (x) = 2x + 1, where, Y = {y ∈ N : y = 4x + 3 for some x ∈ N }.

Show that f is invertible

#### Checking by One-One and Onto Method

Checking one-one

f(x 1 ) = 2x 1 + 1

f(x 2 ) = 2x 2 + 1

Putting f(x 1 ) = f(x 2 )

2x 1 + 1 = 2x 2 + 1

2x 1 = 2x 2

x 1 = x 2

If f(x 1 ) = f(x 2 ) , then x 1 = x 2

∴  f is one-one

Checking onto

f(x) = 2x + 1

Let f(x) = y, where y ∈ Y

y = 2x + 1

y – 1 = 2x

2x =  y – 1

x = (y - 1)/2

For every y in Y = {y ∈ N : y = 2x + 1 for some x ∈ N }.

There is a value of x which is a natural number

Thus, f is onto

Since f is one-one and onto

f is invertible

#### Checking by fog = I Y and gof = I X  method

Checking inverse of f: X → Y

Step 1 : Calculate g: Y → X

Step 2 : Prove gof = I X

Step 3 : Prove fog = I Y

g is the inverse of f

Step 1

f(x) = 2x + 1

Let f(x) = y

y = 2x + 1

y – 1 = 2x

2x =  y – 1

x = (y - 1)/2

Let g(y) = (y - 1)/2

where g: Y → N

Step 2 :

gof = g(f(x))

= g(2x + 1)

= ((2x + 1) - 1)/2

= (2x + 1 - 1)/2

= 2x/2

= x

= I N

Step 3 :

fog = f(g(y))

= f((y - 1)/2)

= 2 ((y - 1)/2) + 1

= y – 1 + 1

= y

= I Y

Since gof   = I N and fog = I Y ,

f is invertible

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