2.4: Composition of Functions

Combining Functions Using Algebraic Operations

Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.

Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year’s incomes and then collecting all the data in a new column. If $w(y)$ is the wife’s income and $h(y)$ is the husband’s income in year $y$, and we want $T$ to represent the total income, then we can define a new function.

$$T(y)=h(y)+w(y) \nonumber$$

If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write

$$T=h+w \nonumber$$

Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.

For two functions $f(x)$ and $g(x)$ with real number outputs, we define new functions $f+g$, $f−g$, $fg$, and $\frac{f}{g}$ by the relations.

$$\begin{align*} (f+g)(x) &=f(x)+g(x) \\[4pt] (f−g)(x) &=f(x)−g(x) \\[4pt] (fg)(x)&=f(x)g(x) \\[4pt] \left(\dfrac{f}{g}\right)(x) &=\dfrac{f(x)}{g(x)} \end{align*}$$

Example $\PageIndex{1}$: Performing Algebraic Operations on Functions

Find and simplify the functions $(g−f)(x)$ and $\left(\dfrac{g}{f}\right)(x)$, given $f(x)=x−1$ and $g(x)=x^2−1$. Are they the same function?

Solution

Begin by writing the general form, and then substitute the given functions.

$$\begin{align*} (g−f)(x) &= g(x)−f(x) \\[4pt] (g−f)(x) &=x^2−1−(x−1) \\[4pt] &=x^2−x \\[4pt] &=x(x−1) \end{align*}$$

$$\begin{align*} \left(\dfrac{g}{f}\right)(x)&=g(x)f(x) \\[4pt] \left(\dfrac{g}{f}\right)(x)&=\dfrac{x^2−1}{x−1} \\[4pt] &=\dfrac{(x+1)(x−1)}{x−1} \\[4pt]&=x+1 \end{align*}$$

No, the functions are not the same.

Note: For $\left(\dfrac{g}{f}\right)(x)$, the condition $x\neq1$ is necessary because when $x=1$, the denominator is equal to 0, which makes the function undefined.

Exercise $\PageIndex{1}$

Find and simplify the functions $(fg)(x)$ and $(f−g)(x)$.

$$f(x)=x−1 \nonumber$$

and

$$g(x)=x^2−1 \nonumber$$

Are they the same function?

Answer

$(fg)(x)=f(x)g(x)=(x−1)(x2−1)=x^3−x^2−x+1 \\[4pt] (f−g)(x)=f(x)−g(x)=(x−1)−(x^2−1)=x−x^2$

No, the functions are not the same.

Create a Function by Composition of Functions

Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:

$$f{\circ}g(x)=f(g(x))$$

We read the left-hand side as“$f$ composed with $g$ at $x$,” and the right-hand side as“$f$ of $g$ of $x$.”The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol $\circ$ is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases $f(g(x)){\neq}f(x)g(x)$.

It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function $g$ takes the input $x$ first and yields an output $g(x)$. Then the function $f$ takes $g(x)$ as an input and yields an output $f(g(x))$.

Figure $\PageIndex{2}$: Explanation of the composite function.

In general, $f{\circ}g$ and $g{\circ}f$ are different functions. In other words, in many cases $f(g(x)){\neq}g(f(x))$ for all $x$. We will also see that sometimes two functions can be composed only in one specific order.

For example, if $f(x)=x^2$ and $g(x)=x+2$, then

$$\begin{align*} f(g(x))&= f(x+2) \\[4pt]&=(x+2)^2 \\[4pt] &=x^2+4x+4 \end{align*}$$

but

$$\begin{align*} g(f(x))&= g(x^2) \\[4pt]&=x^2+2 \end{align*}$$

These expressions are not equal for all values of x, so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value $x=−\frac{1}{2}$.

Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.

Composition of Functions

When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any input $x$ and functions $f$ and $g$, this action defines a composite function, which we write as $f{\circ}g$ such that

$$(f{\circ}g)(x)=f(g(x))$$

The domain of the composite function $f{\circ}g$ is all $x$ such that $x$ is in the domain of $g$ and $g(x)$ is in the domain of $f$.

It is important to realize that the product of functions $fg$ is not the same as the function composition $f(g(x))$, because, in general, $f(x)g(x){\neq}f(g(x))$.

Example $\PageIndex{2}$: Determining whether Composition of Functions is Commutative

Using the functions provided, find $f(g(x))$ and $g(f(x))$. Determine whether the composition of the functions is commutative.

$$f(x)=2x+1 \;\;\;\; g(x)=3−x \nonumber$$

Solution

Let’s begin by substituting $g(x)$ into $f(x)$.

$$\begin{align*} f(g(x))&= 2(3−x)+1 \\[4pt] &=6−2x+1 \\[4pt] &=7−2x \end{align*}$$

Now we can substitute $f(x)$ into $g(x)$.

$$\begin{align*} g(f(x))&= 3−(2x+1) \\[4pt]&=3−2x−1 \\[4pt] &=2-2x \end{align*}$$

We find that $g(f(x)){\neq}f(g(x))$, so the operation of function composition is not commutative.

Example $\PageIndex{3}$: Interpreting Composite Functions

The function $c(s)$ gives the number of calories burned completing $s$ sit-ups, and $s(t)$ gives the number of sit-ups a person can complete in $t$ minutes. Interpret $c(s(3))$.

Solution

The inside expression in the composition is $s(3)$. Because the input to the $s$-function is time, $t=3$ represents 3 minutes, and $s(3)$ is the number of sit-ups completed in 3 minutes.

Using $s(3)$ as the input to the function $c(s)$ gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).

Example $\PageIndex{4}$: Investigating the Order of Function Composition

Suppose $f(x)$ gives miles that can be driven in $x$ hours and $g(y)$ gives the gallons of gas used in driving $y$ miles. Which of these expressions is meaningful: $f(g(y))$ or $g(f(x))$?

Solution

The function $y=f(x)$ is a function whose output is the number of miles driven corresponding to the number of hours driven.

$$\text{number of miles } =f (\text{number of hours}) \nonumber$$

The function $g(y)$ is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:

$$\text{number of gallons } =g(\text{number of miles}) \nonumber$$

The expression $g(y)$ takes miles as the input and a number of gallons as the output. The function $f(x)$ requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression $f(g(y))$ is meaningless.

The expression $f(x)$ takes hours as input and a number of miles driven as the output. The function $g(y)$ requires a number of miles as the input. Using $f(x)$ (miles driven) as an input value for $g(y)$, where gallons of gas depends on miles driven, does make sense. The expression $g(f(x))$ makes sense, and will yield the number of gallons of gas used, $g$, driving a certain number of miles, $f(x)$, in $x$ hours.

Question/Answer

Are there any situations where $f(g(y))$ and $g(f(x))$ would both be meaningful or useful expressions?

Yes. For many pure mathematical functions, both compositions make sense, even though they usually produce different new functions. In real-world problems, functions whose inputs and outputs have the same units also may give compositions that are meaningful in either order

Exercise $\PageIndex{2}$

The gravitational force on a planet a distance $r$ from the sun is given by the function $G(r)$. The acceleration of a planet subjected to any force $F$ is given by the function $a(F)$. Form a meaningful composition of these two functions, and explain what it means.

Answer

A gravitational force is still a force, so $a(G(r))$ makes sense as the acceleration of a planet at a distance $r$ from the Sun (due to gravity), but $G(a(F))$ does not make sense.

Evaluating Composite Functions

Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.

Evaluating Composite Functions Using Tables

When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.

Example $\PageIndex{5}$: Using a Table to Evaluate a Composite Function

Using Table $\PageIndex{1}$, evaluate $f(g(3))$ and $g(f(3))$.

Table $\PageIndex{1}$

$x$	$f(x)$	$g(x)$
1	6	3
2	8	5
3	3	2
4	1	7

Solution

To evaluate $f(g(3))$, we start from the inside with the input value 3. We then evaluate the inside expression $g(3)$ using the table that defines the function $g: g(3)=2$. We can then use that result as the input to the function $f$, so $g(3)$ is replaced by 2 and we get $f(2)$. Then, using the table that defines the function $f$, we find that $f(2)=8$.

$$g(3)=2 \nonumber$$

$$f(g(3))=f(2)=8 \nonumber$$

To evaluate $g(f(3))$, we first evaluate the inside expression $f(3)$ using the first table: $f(3)=3$. Then, using the table for $g$, we can evaluate

$$g(f(3))=g(3)=2 \nonumber$$

Table $\PageIndex{2}$ shows the composite functions $f{\circ}g$ and $g{\circ}f$ as tables.

Table $\PageIndex{2}$

$x$	$g(x)$	$f(g(x))$	$f(x)$	$g(f(x))$
3	2	8	3	2

Exercise $\PageIndex{3}$

Using Table $\PageIndex{1}$, evaluate $f(g(1))$ and $g(f(4))$.

Answer

$f(g(1))=f(3)=3$ and $g(f(4))=g(1)=3$

Evaluating Composite Functions Using Graphs

When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the x- and y-axes of the graphs.

How To ...

Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.

1. Locate the given input to the inner function on the x-axis of its graph.
2. Read off the output of the inner function from the y-axis of its graph.
3. Locate the inner function output on the x-axis of the graph of the outer function.
4. Read the output of the outer function from the y-axis of its graph. This is the output of the composite function.

Example $\PageIndex{6}$: Using a Graph to Evaluate a Composite Function

Using Figure $\PageIndex{3}$, evaluate $f(g(1))$.

Figure $\PageIndex{3}$: Two graphs of a positive and negative parabola.

Solution

To evaluate $f(g(1))$, we start with the inside evaluation. See Figure $\PageIndex{4}$.

Figure $\PageIndex{4}$: Two graphs of a positive parabola $g(x)$ and a negative parabola $f(x)$. The following points are plotted: $g(1)=3$ and $f(3)=6$.

We evaluate $g(1)$ using the graph of $g(x)$, finding the input of 1 on the x-axis and finding the output value of the graph at that input. Here, $g(1)=3$. We use this value as the input to the function $f$.

$$f(g(1))=f(3) \nonumber$$

We can then evaluate the composite function by looking to the graph of $f(x)$, finding the input of 3 on the x-axis and reading the output value of the graph at this input. Here, $f(3)=6$, so $f(g(1))=6$.

Analysis

Figure $\PageIndex{5}$ shows how we can mark the graphs with arrows to trace the path from the input value to the output value.

Figure $\PageIndex{5}$: Two graphs of a positive and negative parabola.

Exercise $\PageIndex{4}$

Using Figure $\PageIndex{3}$, evaluate $g(f(2))$.

Answer

$g(f(2))=g(5)=3$

Evaluating Composite Functions Using Formulas

When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.

While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition $f(g(x))$. To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like $f(t)=t^2−t$, we substitute the value inside the parentheses into the formula wherever we see the input variable.

How To...

Given a formula for a composite function, evaluate the function.

1. Evaluate the inside function using the input value or variable provided.
2. Use the resulting output as the input to the outside function.

Example $\PageIndex{7}$: Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input

Given $f(t)=t^2−t$ and $h(x)=3x+2$, evaluate $f(h(1))$.

Solution

Because the inside expression is $h(1)$, we start by evaluating $h(x)$ at 1.

$$\begin{align*} h(1)=3(1)+2 \\[4pt] h(1)&=5 \end{align*}$$

Then $f(h(1))=f(5)$, so we evaluate $f(t)$ at an input of 5.

$$\begin{align*} f(h(1)) &=f(5) \\[5pt] f(h(1))&=5^2−5 \\[5pt] f(h(1))&=20 \end{align*}$$

Analysis

It makes no difference what the input variables $t$ and $x$ were called in this problem because we evaluated for specific numerical values.

Exercise $\PageIndex{5}$

Given $f(t)=t^2−t$ and $h(x)=3x+2$, evaluate

a. $h(f(2))$
b. $h(f(−2))$

Answer a

8

Answer b

20

Key Equation

• Composite function $(f{\circ}g)(x)=f(g(x))$

Key Concepts

• We can perform algebraic operations on functions. See Example.
• When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
• The function produced by combining two functions is a composite function. See Example and Example.
• The order of function composition must be considered when interpreting the meaning of composite functions. See Example.
• A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
• A composite function can be evaluated from a table. See Example.
• A composite function can be evaluated from a graph. See Example.
• A composite function can be evaluated from a formula. See Example.
• The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See Example and Example.
• Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
• Functions can often be decomposed in more than one way. See Example.

Glossary

composite function

the new function formed by function composition, when the output of one function is used as the input of another