Composite Functions
Travis went shopping for groceries at a supermarket.
He noticed that the discount on different items was dependent on the price range for that item.
He wondered how the two aspects of price and discount were related.
In this case, the price is dependent on another function i.e., cost.
Let us understand how these two are interrelated while solving composite functions.
In this lesson, we will learn how to solve composite functions. We will learn in detail about composite functions examples, composite functions definition, and composite function graph.
Check out the interactive simulation to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.
Lesson Plan
What Is a Composite Function?
Composite Function: Definition
A function that depends on any other function is called a composite function.
A composite function is created by composing one function within another function.
General Form
For any function, say \( \text g(x) \), we infer that "\( \text g\) of \(x\)" is a function in terms of variable \( x \).
Similarly, any function of the form \( \text g ( \text f (x))\) will be read as "\( \text f \) of \( \text g\) of \(x\)".
The function \( \text f ( \text g (x))\) is a composite function here.
\(g\) is the inner function and \(f\) is the outer function.
It can also be represented as:
 \( ( \text f \circ \text g )( x )\)
 \( \text {fg} (x)\)
Domain
To calculate the domain of a composite composition \( ( \text f \circ \text g )( x )\):
 Find the domain of the inside function \( \text g(x) \).
 Find the domain of the outside function \( \text f(x) \).
 Find the inputs \( x \) in the domain of the inside function \( \text g(x) \) for which \( \text g(x) \) lies in the domain of \( \text f(x) \).
 The resulting set of both domains will be the domain of the composite function.
Example
Emily visited an amusement park with her family.
The cost of entry tickets was different for people of different ages.
The three different categories of age groups were kids, adults, and senior citizens.
How can you relate the given information to composite functions?
We take function \(\text f (y)\) = Cost of the ticket
Another function, \( \text g ( x )\) = Age group of the person
Since the cost of the ticket is dependent on the age group of the person, we can say that, \( y = \text g (x) \).
Thus, the composite function to represent the cost of ticket can be written as \(( \text f ( \text g ( x ) ))\).
General form of a composite function = \(( \text f ( \text g ( x )) )\) 
What Is the Symbol for Composite Functions?
A composite function is written in the form: \(( \text f ( \text g ( x ) ))\)
The \(( \circ )\) symbol is used to denote a composite association between two functions.
The expression \( ( \text f \circ \text g )( x )\) means that the function \( \text f\) is dependent on function \( \text g ( x )\) or \( \text g ( x )\) is a function enclosed within function \( \text f\).
Example
The following image illustrates an example of how composite functions are mapped together:
How to Solve Composite Functions?
The composition of two functions can be solved using the following steps:
 Write the composition in another form. The composition written in the form \( (\text f \circ \text g)(x)\) needs to be written as \( \text f ( \text g(x))\).
 For every occurrence of \(x\) in the outside function i.e. \( \text f\), replace \(x\) with the inside function \( \text g(x)\).
 Simplify the answer obtained.
Example
If a function \( \text f (x) = –2x + 7 \) and another function \( \text g (x) = x – 9\), can you find \( (\text f \circ \text g )(x)\)?
\( \text f (x) = –2x + 7 \)
Substitute \( x = \text g(x) = x  9 \)
\( \begin{align*} \text f( \text g(x)) &= 2( x  9) + 7 \\ &= 2x + 25 \end{align*}\)
Composite functions Graph
Let us observe the graphing of different composite functions using the following simulation.
How to Break a Function Into the Composition of Other Functions?
We can break a function into the composition of other functions.
Let us see how this can be done using an example.
Example
\( \text f (x) = (\frac{x  1}{ x})^3\)
The above function can be broken down as a composition of two separate functions,
\( \text f (x) = \text u ( \text v (x)) = (\frac{x  1}{ x})^3 \)
From the above equation, we can deduce that,
\( \begin{align*} \text u (x) &= x^3 \\ \text v (x) &= \frac{ x  1}{ x } \end{align*} \)
 In a composite function, the order of the function is very important because \( ( \text f \circ \text g)(x)\) is not equal to \( ( \text g \circ \text f)(x)\).
 The domain of both functions is important in finding the domain of the resulting composite function.
Solved Examples
Example 1 
Given two functions: \( \text f = {(1, 1), (0, 2), (4, 5)}\) and \( \text g = {(1, 1), (2, 3), (7, 9)}\), find \( ( \text g \circ \text f)\) and determine its domain and range.
Solution
Here,
\( \begin{align*} \text g ( \text f ( x)) &= \text g \circ \text f \\ &= (\text g \circ \text f)( 1 ) \\ &= \text g [ \text f (1) ] \\ &= \text g(1) \\ &= 1 \end{align*}\)
Similarly,
\( \begin{align*} \text g ( \text f ( x)) &= \text g \circ \text f \\ &= (\text g \circ \text f)( 0 ) \\ &= \text g [ \text f (0) ] \\ &= \text g(2) \\ &= 3 \end{align*}\)
Also,
\( \begin{align*} \text g ( \text f ( x)) &= \text g \circ \text f \\ &= (\text g \circ \text f)( 4 ) \\ &= \text g [ \text f (4) ] \\ &= \text g(5) \\ &= \text {undefined} \end{align*}\)
Hence,
\(( \text g \circ \text f ) = {(1, 1), (0, 3)}\)
Therefore, Domain: {1, 0} and Range: {0, 3}
\( \begin{align*} \therefore( \text g \circ \text f ) &= {(1, 1), (0, 3)} \\ \text {Domain} &= (1,0) \\ \text {Range} &= (0, 3) \end{align*} \) 
Example 2 
Casey is solving a problem on composite functions that says two functions \( \text f \) and \( \text g \) are given by:
\( \text f (x) = \sqrt{(x  2)}\)
\( \text g (x) = \ln{(1 + x ^2)} \)
Can you help her find the composite function \( (\text g \circ \text f )( x )\)?
Solution
\( (\text g \circ \text f )( x ) = \text g ( \text f (x))\)
Since \( \text f (x) = \sqrt{(x  2)}\), substituting \( x = \text f (x)\), we get,
\( \begin{align*} \text g ( \text f (x)) &= \ln{( 1+ \text f (x)^2)} \\ &= \ln{( 1 + \sqrt{x  2} ^2)} \\ &= \ln{1 + ( x  2 )} \\ &= \ln{ (x  1)} \end{align*} \)
\(\therefore\) \( (\text g \circ \text f )( x ) = \ln{ (x  1)}\) 
Example 3 
If a function \( \text g (x) = 9 x + 2 \), what would be the result of \( (\text f \circ \text g )(x)\)?
Solution
\( \text g (x) = 9x + 2 \)
Substitute \( x = \text g ( x ) = 9x + 2 \)
\( \begin{align*} \text f( \text g(x)) &= 9( 9x + 2) + 2 \\ &= 81x + 18 + 2 \\ &= 81x + 20 \end{align*}\)
\(\therefore (\text f \circ \text g )(x) = 81x + 20 \) 
Example 4 
Can you help Sam solve the following problem?
\( \text f\) and \( \text g\) are two functions, both defined on the set of real numbers and k is a constant such that,
\( \begin{align*} \text f (x) &= \text k x − 4 \\ \text g (x) &= \text k x + 6 \end{align*}\)
If \( (\text f \circ \text g )(x) = ( \text g \circ \text f)(x)\) for all values of x, what is the value of k?
Solution
Here,
\( \begin{align*} \text f( \text g( x )) &= (k ( \text k x + 6)  4) \\ &= \text k^2 x + 6 \ \text  \ 4 \end{align*}\)
Similarly,
\( \begin{align*} \text g( \text f( x )) &= (k ( \text kx  4) + 6) \\ &= \text k^2x  4 \text k + 6 \end{align*}\)
Since,
\( \begin{align*} \text k ^2 x + 6 \text k 4 &= \text k ^2 x  4 \text k +6 \\ 6 \text k  4 &= 4 \text k + 6 \\ 10 \text k &= 10 \\ \text k &= 1 \end{align*} \)
\(\therefore \text k = 1 \) 

Find the function composition \( \text f \circ \text g \circ \text h\) if \(\begin{align} \text f (x) = \frac{1}{x + 3}\end{align}\) , \(\text g (x) = \sqrt{( x + 2)}\) , \(\text { and } \text h (x) = x^2  2\)
Interactive Questions
Here are a few activities for you to practice.
Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The minilesson targeted the fascinating concept of composite functions. The math journey around composite fuctions starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.
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Frequently Asked Questions (FAQs)
1. Are composite functions commutative?
No, composite functions are not commutative because \( ( \text f \circ \text g)(x)\) is not equal to \( ( \text g \circ \text f)(x)\)
2. How to find the domain of a composite function?
To calculate the domain of a composite composition \( ( \text f \circ \text g )( x )\):
 Find the domain of the inside function \( \text g(x) \).
 Find the domain of the outside function \( \text f(x) \).
 Find the inputs \(x\) in the domain of the inside function \( \text g(x) \) for which \( \text g(x) \) lies in the domain of \( \text f(x) \).
 The resulting set of both domains will be the domain of the composite function.
3. How to find the square root of a composite function?
The square root of a composite function can be calculated simply by taking square root as another outside function \(\text f\) and the given composite function \( \text g\) as the inside function.
Thus, solve the obtained \( \text f ( \text g ( x ) )\) function to calculate the square root.
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