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

1.	Show that the function f : R → R : f(x) = 2x + 3 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= x₂or 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 ∈ R and f(x) = 2x+3.So f(x₁) = f(x₂) → 2x₁+3 = 2x²+3 → x₁=x₂ So f(x₁) = f(x₂) ↔ x₁= x₂, f(x) is one-one Given co-domain of f(x) is R. Let y = f(x) = 2x+3 , So x = \(\frac{y-3}{2}\) [Range of f(x) = Domain of y] So Domain of y is R(real no.) = Range of f(x) Hence, Range of f(x) = co-domain of f(x) = R So, f(x) is onto function As it is bijective function. So it is invertible Invers of f(x) is f^-1(y) = \(\frac{y-3}{2}\)

Show that the function f : R → R : f(x) = 2x + 3 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= x₂or 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 ∈ R and f(x) = 2x+3.So f(x₁) = f(x₂) → 2x₁+3 = 2x²+3 → x₁=x₂

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

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

Let y = f(x) = 2x+3 , So x = \(\frac{y-3}{2}\) [Range of f(x) = Domain of y]

So Domain of y is R(real no.) = Range of f(x)

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

So, f(x) is onto function

As it is bijective function. So it is invertible

Invers of f(x) is f^-1(y) = \(\frac{y-3}{2}\)