Composition of function

Composition of Functions 

Composition of functions is a mathematical operation that involves combining two functions to form a new function. The output of one function is used as the input of the other function, and the resulting function is a combination of the two.

The composition of two functions f and g, denoted by f o g, is defined as:

(f o g)(x) = f(g(x))

In other words, the output of g(x) is plugged into f(x), so the composition of the two functions is the new function that results.

Here's an example of how to compute the composition of two functions:

Let f(x) = x2 and g(x) = x + 1. Then the composition of f and g is given by:

(f o g)(x) = f(g(x)) = f(x + 1) = (x + 1)2 = x2 + 2x + 1

x→ g(x)→ f(g(x)) → (f o g)(x)

 

Properties of Composition of Function

 (i) Let f : A → B and g : B → C be two functions. Then, the composition of f and g, denoted by g o f, is defined as the function g o f : A → C given by g o f (x) = g (f (x)), ∀ x ∈ A.


(ii) If f : A → B and g : B → C are one-one, then g o f : A → C is also one-one


(iii) If f : A → B and g : B → C are onto, then g o f : A → C is also onto.
However, converse of above stated results (ii) and (iii) need not be true. Moreover, we have the following results in this direction.


(iv) Let f : A → B and g : B → C be the given functions such that g o f is one-one. Then f is one-one.


(v) Let f : A → B and g : B → C be the given functions such that g o f is onto. Then g is onto.