Chapter-3 | Composite Functions | Class-10

Composite Function

If A,B, and C are three non-empty sets and f:A->B and g:B->C be the functions then the composite function of f and g denoted by God or gf is denoted from A to C. 

Mathematically, it is defined as (got):A->C: (got)(x) = g[f(x)] for all x of A.

Important tips for Mapping a Composite Function:



When we have two or more than two functions, and we try to link the relation of one function to the another, we receive a new function. This is said to be composite function. 



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Combination of Function and Composite Functions:

Combination of Function:
Applying the mathematical operations on the functions like: addition, subtraction, division and multiplication can be done easily.

We can either combine and evaluate or evaluate and combine to get the sum, difference, product or quotient.

Let us say, we have: f(x) = 2x and g(x) = x+7

To evaluate: f(x) + g(x) at x= 4,

We can, evaluate then combine:
Evaluating, f(4) = 2.4 = 8
g(4) = 4+7 = 11
Now, combine: f(x) + g(x) = 8+11 = 19

Combine then evaluate:
Combining, f(x) + g(x) = 2x +x+7
= 3x +7
Now, evaluate: f(4) + g(4) = 3.4 +7 
= 12+7
= 19
So, whatever method we use, our answer will always be the same. This is applicable to subtraction, multiplication and division too.

Composition of Functions:

Composition of Functions means to apply one function to another. A small circle (o) is used to represent composition of Functions. 

Let us consider, we have fog(x). It would be 'f composed with g of x'. The function outside should be written first.

Let, f(x) = 2x
g(x) = x+6
So, fog(x) = f[g(x)]
= f [x+6]
= 2 [x+6]
= 2x +12
Things to remember:
The outside function should always be written first.
The value of inside function should be kept first.
The value of the inside function holds the value of 'x' of the outside function.

Understanding literally, In fog(x);
g(x) should be in the domain of f(x). 

Decomposition of Functions:

Breaking the already composed functions into two functions is called decomposition of functions.

Like in composition of functions, we applied one function to another but in decomposition, we will try to find the root functions that formed the composed functions.

In other words, if fog(x) is a composite function. We will find f(x) and g(x) in decomposition of functions. 

There are several ways of decomposing the same type of function. Therefore, you can find lots of answers to the same decomposition of functions. 

Here is a small example. If fog(x) = (8x), decompose the function.
Solution:
Let f(x) = x
g(x) = 8x
fog(x) = f[g(x)] = f[8x] = 8x

Also,
Let, 
f(x) = 2x
g(x) = 4x
fog(x) = f[g(x) = f[4x] = 2[4x] = 8x

Again,
Let,
f(x) = 16x
g(x) = 1/2x
fog(x) = f[g(x) = f[1/2x] = 16*1/2x = 8x

There are other several ways of decomposing this same function. Therefore, every individual may have unique answer to this question.