Show that the function f: R → R: f(x) = 2x + 3 is invertible and fin

1.	Show that the function f: R → R: f(x) = 2x + 3 is invertible and find f -1.
Answer» We know that f(x₁) = f(x₂) It can be written as 2x₁ + 3 = 2x₂ + 3 On further calculation 2x₁ = 2x₂ So we get x₁ = x₂ Hence, f is one-one. Consider y = 2x + 3 It can be written as y – 3 = 2x So we get x = (y – 3)/ 2 If y ∈ R, there exists x = (y – 3)/ 2 ∈ R f (x) = f ([y-3]/2) = 2([y – 3]/ 2) +3 = y f is onto Here, f is one-one onto and invertible. Take y = f(x) It can be written as y = 2x + 3 So we get x = (y-3)/ 2 So f ^-1 (y) = (y – 3)/ 2 Hence, we define f ^-1: R → R: f ^-1(y) = (y – 3)/ 2 for all y ∈ R

Show that the function f: R → R: f(x) = 2x + 3 is invertible and find f -1.

Answer»

We know that

f(x₁) = f(x₂)

It can be written as

2x₁ + 3 = 2x₂ + 3

On further calculation

2x₁ = 2x₂

So we get

x₁ = x₂

Hence, f is one-one.

Consider y = 2x + 3

It can be written as

y – 3 = 2x

So we get

x = (y – 3)/ 2

If y ∈ R, there exists x = (y – 3)/ 2 ∈ R

f (x) = f ([y-3]/2) = 2([y – 3]/ 2) +3 = y

f is onto

Here, f is one-one onto and invertible.

Take y = f(x)

It can be written as

y = 2x + 3

So we get

x = (y-3)/ 2

So f ^-1 (y) = (y – 3)/ 2

Hence, we define f ^-1: R → R: f ^-1(y) = (y – 3)/ 2 for all y ∈ R