Composition of functions | Invertible function

Invertible Function

A function f: X → Y is said to be invertible if there exists another function g: Y → X such that for every element y ∈ Y, g(y) = x where x ∈ X.

In this case, g is called the inverse of f and is denoted by f⁻¹. The following properties must hold:

• g(f(x)) = I_X(x) = x for all x ∈ X
• f(g(y)) = I_Y(y) = y for all y ∈ Y

Here, I_X and I_Y represent the identity functions on sets X and Y respectively.

Conditions for Invertibility

A function f is invertible if and only if:

• Injective (one-to-one): Each distinct input in the domain corresponds to a unique output in the codomain.
• Surjective (onto): Every value in the codomain has at least one pre-image in the domain.

Therefore, only bijective functions (both injective and surjective) have inverses.

Example 1:

Let f: ℝ → ℝ be defined by f(x) = x + 3.

To find the inverse:

y = f(x) = x + 3  
=> x = y - 3  
So, f⁻¹(y) = y - 3
  

Example 2:

Let f(x) = 2x, where f: ℝ → ℝ

y = f(x) = 2x  
=> x = y / 2  
So, f⁻¹(y) = y / 2
  

Graphical Interpretation:

If you graph a function and its inverse, they are mirror images across the line y = x.