Composite Functions

The flour of the bread can be referred to by x, and the function f(x) is the machine for making bread, and g(x) is the function when the bread is baked in the oven. The prepared bread which comes as output is referred to as f(g(x)). This output is known as the composition of functions. This article will explain the composite functions in a structured and ordered way.

What are Composite functions?

A composite function is formed when one function relates to another. For example, f(g(x)). In f(g(x)), the first function is f(x) and the second function is g(x). This is a combination of two functions which ultimately results in a single function.

Types of Functions

The g(x) is the input of the function f(g(x)). f(g(x)) can be also written as fog(x).
The f(x) is the input of the function g(f(x)). g(f(x)) can be also written as gof(x).

Composite Functions – Symbols

The composite functions can be represented with the symbol o. This symbol can be ignored and in its place brackets can be used to represent composite functions. The relevant examples are given below:

fog(x): In this case, g(x) is the inner function and f(x) is the outer function. This can also be written as f(g(x)).
gof(x): In this case, f(x) is the inner function and g(x) is the outer function. This can also be written as g(f(x)).

Step by Step Process to Solve Composite Functions

Composite functions are tough to solve at one glance. However, this problem can be solved through a process which is discussed as follows:

The BODMAS principle of mathematics stands for brackets of division, multiplication, addition, and subtraction in mathematics. It can also be applied in composite functions.
First, you need to clear the brackets starting from the inner function. For the calculation of f(g(x)), you need to substitute the value of x into g(x), and then substitute the value of g(x) into f(x).
Whenever you are calculating g(f(x)), then substitute the value of x in f(x), then substitute the value of f(x) into g(x). You will get the result after this process.

Keep one thing in your mind that it’s not necessary for fog(x) to be equal to gof(x).

The domain of the Composite Functions

To find the domain of the composite functions, follow the process mentioned below:

First, you need to calculate the domain of the function g(x). Name this function as A.
Then, you need to calculate the domain of another function f(g(x)). Let us name this function B.
After this, you need to find the intersection of the two functions A and B. This will help you to find the domain of f(g(x)).

The range of composite functions

Calculating the range of composite functions, follows the same process as normal functions. The range of the composite function does not depend upon the inner and outer functions.