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» To Show: that f is invertible To Find: Inverse of f [NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)] one-one function: A function f : A B is said to be a one-one function or injective mapping if different elements of A have different images in B. Thus for x₁, x₂ ∈ A & f(x₁), f(x₂) ∈ B, f(x₁) = f(x₂) ↔ x₁= x₂ or x₁ ≠ x₂ ↔ f(x₁) ≠ f(x₂) onto function: If range = co-domain then f(x) is onto functions. So, We need to prove that the given function is one-one and onto. Let x₁, x₂ ∈ Q and f(x) = 3x-4.So f(x₁) = f(x₂) → 3x₁ - 4 = 3x₂ - 4 → x₁=x₂ So f(x₁) = f(x₂) ↔ x₁= x₂, f(x) is one-one Given co-domain of f(x) is Q. Let y = f(x) = 3x- 4 , So x = [Range of f(x) = Domain of y] So Domain of y is Q = Range of f(x) Hence, Range of f(x) = co-domain of f(x) = Q So, f(x) is onto function As it is bijective function. So it is invertible Invers of f(x) is f -1(y) =\(\frac{y+4}{3}\)

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

Answer»

To Show: that f is invertible

To Find: Inverse of f

[NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)]

one-one function: A function f : A B is said to be a one-one function or injective mapping if different

elements of A have different images in B. Thus for x₁, x₂ ∈ A & f(x₁), f(x₂) ∈ B, f(x₁) = f(x₂) ↔ x₁= x₂ or x₁ ≠ x₂ ↔ f(x₁) ≠ f(x₂)

onto function: If range = co-domain then f(x) is onto functions.

So, We need to prove that the given function is one-one and onto.

Let x₁, x₂ ∈ Q and f(x) = 3x-4.So f(x₁) = f(x₂) → 3x₁ - 4 = 3x₂ - 4 → x₁=x₂

So f(x₁) = f(x₂) ↔ x₁= x₂, f(x) is one-one

Given co-domain of f(x) is Q.

Let y = f(x) = 3x- 4 , So x = [Range of f(x) = Domain of y]

So Domain of y is Q = Range of f(x)

Hence, Range of f(x) = co-domain of f(x) = Q

So, f(x) is onto function

As it is bijective function. So it is invertible

Invers of f(x) is f -1(y) =\(\frac{y+4}{3}\)