Composition of Functions

In Mathematics, generally, we refer to a function as a rule that describes and relates a given set of inputs to a set of relevant possible outputs. The important and key point to remark regarding a function is that every input is usually related to exactly one output. The method of naming certain functions is known as function notation. The most regularly employed function notation symbols incorporate $f(x),~g(x),~h(x),...$ etc.

In this article, we will focus on a special type of function known as the composite function and will learn the concepts behind the composition of functions, what is a composite function, alongside the properties of composite functions.

Composite Functions:

Generally in Mathematics, the composition of functions or the composite functions are the operation where two functions say $f(x)$ and $g(x)$ results in a new function say $h(x)$ in such a way that $h(x)=g(f(x))$. It implies that the function $g(x)$ is applied to the function of $x$. So, in other words, it means that a function is employed to the result of another function.

What is a Composite Function?

Now, let us have a look at what is a composite function? Suppose that we have two different functions, such that we can generate another new function by composing one function into the other. The steps needed to execute this operation are similar to when any function is solved for any given value. Such functions are known as composite functions.

Thus, a composite function is defined as a function, which is written within another function. Hence, the composition of functions is generated by substituting one function within another function.

Let’s understand these compositions of functions with a suitable example. Let’s say that the function $f(g(x))$ is the composite function of two different functions$f(x)$ and $g(x)$. The function $g(x)$ is known as an inner function and the function $f(x)$ is referred to as an outer function. Hence, it is also read $f(g(x))$ as “the function $g(x)$ is the inner function of the outer function $f(x)$”.

General Form of Composite Functions

• $f(x)$ is known as an inner function and $g$ as an outer function.

• The composite functions are generally denoted by the symbol $o$ And the general form of composite functions or the composition of functions formula is written as:

$\Rightarrow(fog)x$ or $f(g(x))$

• One of the important points to be noted is here that any composite function says $f(g(x))$ will not be the same as the $g(f(x))$ i.e., $(fog)\neq(gof)$.

Domain of The Composite Functions

• The domain of any composite function is the set of values that goes into the function. For example, for the composite function $f(g(x))$, $g(x)$ is the domain of function $f$.

• To determine the value of the domain of any composition of functions $(fog)$ Initially, determine the domain of the inside function here it is $g(x)$.

• Find the domain of the outside function $f(x)$ Find the inputs $x$ in the domain of the inside function $g(x)$ such that the function $g(x)$ lies within the domain of $f(x)$.

• Finally obtained set of both domains will be the domain of the considered composite function $(fog)$.

Composite Functions Properties:

• The composite functions of one to one functions are always one to one.

• The composite functions of two onto functions are always onto.

• The inverse of the two composite functions f and g is equal to the composition of the inverse of both the functions i.e., $(fog)^{-1}=(g^{-1}of^{-1})$.

• Composition of functions is associative: The composite functions f, g and hare said to be associative in nature if and only if $fo(goh)=(fog)oh$

How to Solve Composition of Functions

The composition of two functions can be solved using the following steps:

• Note down the composition of functions in another form. The composition written in the form $(fog)$ needs to be written as $f(g(x))$.

• For every appearance of $g$ in the outside function i.e. $g$, replace with the inside function $g(x)$. Finally, just go for simplification of the answer obtained.

Composition of Functions Examples:

1. Given that two functions $f(x)=4x^2-x$ and $g(x)=x+1$ Evaluate $(fog)x$ and hence show that $(fog)\neq(gof)$.

Sol:

Given,

$f(x)=4x^2-x$

$g(x)=x+1$

Now, we are asked to evaluate the composite function $(fog)x$ We know that in order to evaluate any composite function we should follow the composite function definition. The composite function can be solved by replacing the input values of the domain of the outer function with the inner function.

Thus, replace the value of $x$ by $g(x)$ in the function $f(x)$ Thus we get:

$f(g(x))=4(g(x))^2-7(g(x))~~~~~...(1)$

Substituting the value of $g(x)$ in the above expression (1) and simplify. We get:

$\Rightarrow (fog)x=4(x+1)^2-7(x+1)$

$\Rightarrow (fog)x=4(x^2+2x+1)-7x-7$

$\Rightarrow (fog)x=4x^2+8x+4-7x-7$

$\Rightarrow (fog)x=4x^2+x-3~~~~~...(2)$

Thus the composite function of the given two functions is $4x^2+x-3$

Now, our aim is to show that $(fog)\neq(gof)$ Thus, let us calculate $(gof)x$ We know that the given composite function is evaluated by replacing the input values of the domain of the outer function with the inner function.

Thus, replace the value of $x$ by $f(x)$ in the function $g(x)$ Thus we get:

$\Rightarrow (gof)x=g(f(x))=f(x)+1$

$\Rightarrow (gof)x=4x^2-7x+1~~~~~...(3)$

From equations (2) and (3), it is proven that $(fog)\neq(gof)$

2. Given $f(x)=6x+4$, find $(fof)x$

Sol: Given, $f(x)=6x+4$

Now, we are asked to evaluate the composite function $(fof)x$ We know that in order to calculate any composite function we should follow the definition of the composite functions. The composition of functions can be found by replacing the input values of the domain of the outer function with the inner function.

Here, both inner and outer functions are $f(x)$

Thus, replace the value of $x$ by $f(x)$ in the function$f(x)$. Thus we get:

$\Rightarrow (fof)x=(f(x))=6(f(x))+4$

$\Rightarrow (fof)x=6(6x+4)+4$

$\Rightarrow (fof)x=36x+24+4$

$\Rightarrow (fof)x=36x+28$

Therefore, the composite function $(fof)x$ is $36x+28$.

Practise Question MCQs

1. Suppose that $f(x)=4x^2-1$ and $g(x)=1-x$, which of the following shows the expression for $(fog)x$?

1. 2-4x2

2. 4x2-2

3. 4x2-8x-3

4. 4x2-8x+3

Ans: D

2. Which of the following shows the composition function when the inner function is a square root function, $y=\sqrt{3x-2}$ , and the outer function is given by y=2x2−1?

1. 3x-3

2. 6x-5

3. 3x+3

4. 6x+5

Ans: B

Conclusion

A function that relies on the results of any other function is known as a composite function. A composite function is generated by composing one function within another function. In this article, we have come across problems and practise questions for better understanding.