3. Inverse and Composition of Functions

 

Identity Functions

The function ƒ is called identity function, if each element of set A has an image on itself i.e. ƒ(a) = a ∀ a ∈ A. It is denoted by I. If a function is an identity function, then it will be one-to-one as well as onto function or simply a bijective function.

Invertible (Inverse or Anti-Function) Function

A function that reverse another function. A function ƒ : A → B is invertible if and only if it is a bijective function. The inverse function for ƒ exists if ƒ^-1 is a function from B → A.

In the function ƒ applied to an input x gives a result of y, then applying its inverse function ƒ^-1 to y gives the result x, and vice-versa.

ƒ(x) = y then ƒ^-1(y) = x

Composition of Functions

Function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)).

In this operation, the function g is applied to the result of applying the function f to x and is denoted as gof(x). That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z.

fog = f(g(x))
gof = g(f(x))

The composition of functions is always associative. That is, if f, g and h are three functions with suitably chosen domains and co-domains, then f ⚬ (g ⚬ h) = (f ⚬ g) ⚬ h.

Example :

If f = {(1, 3), (2, 1), (3, 4), (4, 6), (5, 5), (6, 2)} and
g = {(1, 5), (2, 3), (3, 4), (4, 1), (5, 6), (6, 2)}.

Then composition of function is
gof = g ⚬ f = {(1, 4), (2, 5), (3, 1), (4, 2), (5, 6), (6, 3)}

fog = f ⚬ g = {(1, 5), (2, 4), (3, 6), (4, 3), (5, 2), (6, 1)}

Dirichlet's Drawer Principle

Also known as Pigeon hole principle or shoe box arguments. Let A and B be any two non-empty finite sets. If |A| > |B|, then for functions from A to B, there exists a₁, a₂ ∈ Domain such that ƒ(a₁) = ƒ(a₂).

Here, we have |A| = 5 and |B| = 3. As the Dirichlet's Drawer Principles states, here we have ƒ(1) = ƒ(4) and ƒ(2) = ƒ(5).