# Inverse of a Function

## Inverse of a Function:

Consider a one-one onto function, f: x → y and let y be any arbitrary element of Y. Since f is onto, there exists at least one element x ∈ X such that f(x) = y. Again, as f is one-one, the element x ∈ X is unique. So corresponding to every element y ∈ Y, there exists a unique element x ∈ X and we can define a function g from Y to X i.e., g: Y → X such that g(y) = x for all y ∈ Y. This function g, associated with the function f, is known as the inverse of f and is denoted by f-1.

Thus, if f: X → Y be one-one onto function and if f(x) = y, where x ∈ X and y ∈ Y, then f-1: Y → X defined by f-1(y) = x is called the inverse of the function f(x).

A function whose inverse exists is known as an invertible or an inversible function. Obviously, the domain of f-1 = range of f.

Properties of Invertible Functions:

• The inverse of a bijection is unique.
• The inverse of a bijection is also a bijection.
• If f: A → B and g: B → C are two bijections, then gof: A → C is a bijection, and (gof)-1 = f-1og-1.
• If f: A → B is a bijection and g: B → A is the inverse of f, then fog = IB and gof = IA, where IA and IB are identity functions on the sets A and B respectively.
• If f: A → B and g: B → A be two functions such that gof = IA and fog = IB, then f and g are bijections and g = f-1.