Question

Let A = R – {2} and B = R – {1}. If f: A `→` B is a function defined by f(x) = `(x - 1)/(x - 2)` then show that f is a one-one and an onto function.

Answer 1

Let A = R – {2}, B = R – {1}

f: A `→` B s.t. f(x) = `(x - 1)/(x - 2)`

For x₁, x₂ ∈ A

f(x₁) = f(x₂)

`\implies (x_1 - 1)/(x_1 - 2) = (x_2 - 1)/(x_2 - 2)` 

`\implies (x_1 - 1)/(x_1 - 2) - 1 = (x_2 - 1)/(x_2 - 2) - 1`

`\implies 1/(x_1 - 2) = 1/(x_2 - 2)`

`\implies` x₁ – 2 = x₂ – 2

`\implies` x₁ = x₂

∴ f(x) is one-one function.

Also if f(x) = y, where y ∈ B.

`\implies (x - 1)/(x - 2)` = y 

`\implies` x – 1 = xy – 2y

`\implies` 2y – 1 = xy – x

`\implies` 2y – 1 = x(y – 1)

x = `(2y - 1)/(y - 1) ∈ A`

Clearly every element y ∈ B is associated to x = `(2y - 1)/(y - 1)` of set A.

So Range of f = B `\implies` f is into

Hence f is one-one and onto function.