Question

Let ݂f : A → B be a function defined as f(x) = (2x + 3)/(x - 3), where A = R − {3} and B = R − {2}. Is the function f one – one and onto? Is f invertible? If yes, then find its inverse.

Answer 1

Now f(x₁) = f(x₂)

= x₁ = x₂

so f(x) is one - one 

For onto

x = 3(y + 1)/(y - 2).......(2)

equation (2) is defined for all real values of y except 2 

i.e y ∈ R − { 2 } which is same as given set B = R − { 2 } 

(co-domain = range)

Also y = f(x)

f(x) = f{3(y + 1)/(y - 2)}

(since f(x) = (2x + 3)/(x - 3)

2(3y + 3) + 3(y - 2)/ 3y + 3 - 3y + 6 = 9y/9 = y

Thus for every y ∈ B, there exists x ∈ A such that f ( x ) = y

Thus function is onto. Since f ( x ) is one -one and onto so f ( x ) is invertible.

Inverse is given by x = f ^-1(y) = 3(y + 1)/(y - 2)