Composition of Functions: Overview, Questions, Preparation

The function composition is a function where two functions say that f and g generate a new function, say h, in such a way that h(x) = g(f(x)). This implies that the function g is extended to function x. So, essentially, a function is added to the output of a different function.  This composition of Function is important topic for Relations and Functions Class 12 Maths of the NERT Textbook and subsequently CBSE Boards.

Symbol: It is often denoted as (goh)(x), where there is a small circle symbol. It cannot be substituted by a dot (.) since it would be the output of two functions, such as (g.f) (x). 

Domain: f(g(x)) shall be interpreted as f of g of x. In the composition of (f o g) (x) the function domain f becomes g (x). The domain is set to all values that are used in the function. 

Properties of the Composition of Functions

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

Commutative Property: two functions f and g are said to be swapped with each other, if and only if; 
g_￮f = f_￮g
A few more of the properties are: 

• The one-to-one feature composition is still one-to-one. 
• The two-on-function composition is still on 
• The reciprocal of the composition of the two variables f and g is equivalent to the composition of the inverse of the two functions, such as (f￮g)-1=(g-1￮ f-1)

Composition of Function With Itself 

It is necessary to compose a function on its own. Suppose f is a function, so the composition of the function f will be on its own. 
(f_￮f)(x) = f(f(x))

This concept is taught under the chapter Relations and Functions. You will learn about the different functions and their properties. The weightage of this topic is 8 marks in the final exam.

1. f(x) = 2x +1 and g(x) = -x^2, then find (g∘f)(x) for x = 2.

Solution:

Given,
f(x) = 2x+1
g(x) = -x^2
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

2. If f(x) = x – 3 and g(x) = 4x² – 3x – 9, find(gof)(x)    
Solution:

(g o f )(x) = g(f (x)) 
= 4(x — 3)²— 3(x — 3) — 9 
=4(x² — 6x +9)— 3(x — 3)— 9 
= 4x²— 24x +36-3x +9-9 
= 4x²— 27x +36 

Q: How can you figure out the composition of a function? 

Q: Is the composition of an associative function? 

Q: What are the features of the function composition? 

Q: What is the composition of the function examples? 

Q: How can you create a composite function?