How to check if function has inverse?

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