Let f: Q → Q: f(x) = 3x – 4. Show that f is invertible and find f�

1.	Let f: Q → Q: f(x) = 3x – 4. Show that f is invertible and find f -1.
Answer» We know that f(x₁) = f(x₂) It can be written as 3x₁ – 4 = 3x₂ – 4 So we get 3x₁ = 3x₂ Where x₁ = x₂ f is one-one. Take y = 3x – 4 It can be written as y + 4 = 3x So we get x = (y + 4)/ 3 If y ∈ R there exists x = (y + 4)/ 3 ∈ R We know that f (x) = f([y + 4]/ 3) = 3 ([y + 4]/ 3) – 4 = y f is onto Here, f is one-one onto and invertible. Take y = f(x) So we get y = 4x – 3 It can be written as x = (y + 4)/ 3 So f ^-1 (y) = (y + 4)/ 3 Hence, we define f ^-1: R → R: f ^-1(y) = (y + 4)/ 3 for all y ∈ R

Let f: Q → Q: f(x) = 3x – 4. Show that f is invertible and find f -1.

Answer»

We know that

f(x₁) = f(x₂)

It can be written as

3x₁ – 4 = 3x₂ – 4

So we get

3x₁ = 3x₂

Where x₁ = x₂

f is one-one.

Take y = 3x – 4

It can be written as

y + 4 = 3x

So we get

x = (y + 4)/ 3

If y ∈ R there exists x = (y + 4)/ 3 ∈ R

We know that f (x) = f([y + 4]/ 3) = 3 ([y + 4]/ 3) – 4 = y

f is onto

Here, f is one-one onto and invertible.

Take y = f(x)

So we get

y = 4x – 3

It can be written as

x = (y + 4)/ 3

So f ^-1 (y) = (y + 4)/ 3

Hence, we define f ^-1: R → R: f ^-1(y) = (y + 4)/ 3 for all y ∈ R