 # Composition of Functions

In Maths, the composition of function is an operation where two functions say f and g generate a new function say h in such a way that h(x) = g(f(x)). It means here function g is applied to the function of x. So, basically, a function is applied to the result of another function. The relation and function is an important concept of Class 11 and 12. See below the function composition symbol and domain with example.

Symbol: It is also denoted as (g∘f)(x), where ∘ is a small circle symbol. We cannot replace ∘ with a dot (.), because it will show as the product of two functions, such as (g.f)(x).

Domain: f(g(x)) is read as f of g of x. In the composition of (f o g) (x) the domain of function f becomes g(x). The domain is set of all values which go into the function.

Example: If f(x) = 3x+1 and g(x) = x2 , then f of g of x, f(g(x)) = f(x2) = 3x2+1.

If we reverse the function operation, such as f of f of x, g(f(x)) = g(3x+1) = (3x+1)2

## Properties of Function Compositions

Associative Property: As per the associative property of function composition, if there are three functions f, g and h, then they are said to be associative if and only if;

f ∘ (g ∘ h) = (f ∘ g) ∘ h

Commutative Property: Two functions f and g are said to be commute with each other, if and only if;

g ∘ f = f ∘ g

Few more properties are:

• The function composition of one-to-one function is always one to one.
• The function composition of two onto function is always onto
• The inverse of the composition of two functions f and g is equal to the composition of the inverse of both the functions, such as (f ∘ g)-1 = ( g-1 ∘ f-1).

### Function Composition With Itself

It is possible to compose a function with itself. Suppose f is a function, then the composition of function f with itself will be

(f∘f)(x) = f(f(x))

Let us understand this with an example:

Example: If f(x) = 3x2, then find (f∘f)(x).

Solution: Given: f(x) = 3x2

(f∘f)(x) = f(f(x))

= f (3x2)

= 3(3x)2

= 3.9x2

= 27x2

### Example of Composition of Functions

Q.1: If f (x) = 2x and g(x) = x+1, then find (f∘g)(x) if x = 1.

Solution: Given, f(x) = 2x

g(x) = x+ 1

Therefore, the composition of f from g will be;

(f∘g)(x) = f(g(x)) = f(x+1) = 2(x+1)

Now putting the value of x = 1

f(g(1)) = 2(1+1) = 2 (2) = 4

Q.2: If f(x) = 2x +1 and g(x) = -x2, then find (g∘f)(x) for x = 2.

Solution: Given,

f(x) = 2x+1

g(x) = -x2

To find: g(f(x))

g(f(x)) = g(2x+1) = -(2x+1)2

Now put x =2 to get;

g(f(2)) = -(2.2+1)2

= -(4+1)2

=-(5)2

=-25

Q.3: If there are three functions, such as f(x) = x, g(x) = 2x and h(x) = 3x. Then find the composition of these functions such as [f ∘ (g ∘ h)] (x) for x = -1.

Solution: Given,

f(x) = x

g(x) = 2x

h(x) = 3x

To find: [f ∘ (g ∘ h)] (x)

[f ∘ (g ∘ h)] (x) = f ∘ (g(h(x)))

= f ∘ g(3x)

= f(2(3x))

= f(6x)

= 6x

If x = -1, then;

[f ∘ (g ∘ h)] (-1) = 6(-1) = -6