Let be a function defined as f(x) = , where and . Is the function one-one or onto? Is f invertible? If yes, then find its inverse.

Given: f(x) = where and and

To Find: Function f is invertible or not. If it is, then its inverse.

Here, f(x) =

Checking one-one:

Let, x₁, x₂ A

f(x₁) = and f(x₂) =

Equating f(x₁) and f(x₂) we get,

⇒

⇒ x₁ = x₂

So, the function is one-one.

Checking onto:

Let x A and y B

y = f(x) =

⇒ y (x – 3) = 2x + 3

⇒ xy – 3y = 2x + 3

⇒ x (y – 2) = 3(y + 1)

⇒ x =

For y = 2, the value of x is undefined.

But here, , it means B doesn’t contain 2.

Putting f(x) in place of x we get,

f(x) =

So, for every y B there exists a x A

So, the function is onto.

As the function is both, one-one and onto, the function f(x) in invertible. And inverse of the function is