Functions

# Composition of Functions

### Learning Objectives

In this section, you will:

• Combine functions using algebraic operations.
• Create a new function by composition of functions.
• Evaluate composite functions.
• Find the domain of a composite function.
• Decompose a composite function into its component functions.

Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.

Using descriptive variables, we can notate these two functions. The functiongives the costof heating a house for a given average daily temperature indegrees Celsius. The functiongives the average daily temperature on dayof the year. For any given day,means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperatureFor example, we could evaluateto determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function at that temperature. We would write

By combining these two relationships into one function, we have performed function composition, which is the focus of this section.

### 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. Ifis the wife’s income andis the husband’s income in yearand we wantto represent the total income, then we can define a new function.

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

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 functionsandwith real number outputs, we define new functionsandby the relations

### Performing Algebraic Operations on Functions

Find and simplify the functionsandgivenandAre they the same function?

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

No, the functions are not the same.

Note: Forthe conditionis necessary because whenthe denominator is equal to 0, which makes the function undefined.[/hidden-answer]

### Try It

Find and simplify the functionsand

Are they the same function?

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:

We read the left-hand side ascomposed withat and the right-hand side asofofThe two sides of the equation have the same mathematical meaning and are equal. The open circle symbolis 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

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 functiontakes the inputfirst and yields an outputThen the functiontakesas an input and yields an output

In general,andare different functions. In other words, in many casesfor allWe will also see that sometimes two functions can be composed only in one specific order.

For example, ifandthen

but

These expressions are not equal for all values ofso the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value

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 inputand functionsandthis action defines a composite function, which we write assuch that

The domain of the composite functionis allsuch thatis in the domain ofandis in the domain of

It is important to realize that the product of functionsis not the same as the function compositionbecause, in general,

### Determining whether Composition of Functions is Commutative

Using the functions provided, findandDetermine whether the composition of the functions is commutative.

Let’s begin by substitutinginto

Now we can substituteinto

We find thatso the operation of function composition is not commutative.

### Interpreting Composite Functions

The functiongives the number of calories burned completingsit-ups, andgives the number of sit-ups a person can complete inminutes. Interpret

The inside expression in the composition isBecause the input to the s-function is time,represents 3 minutes, andis the number of sit-ups completed in 3 minutes.

Usingas the input to the functiongives 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).

### Investigating the Order of Function Composition

Supposegives miles that can be driven inhours andgives the gallons of gas used in drivingmiles. Which of these expressions is meaningful:or

The functionis a function whose output is the number of miles driven corresponding to the number of hours driven.

The functionis a function whose output is the number of gallons used corresponding to the number of miles driven. This means:

The expressiontakes miles as the input and a number of gallons as the output. The functionrequires a number of hours as the input. Trying to input a number of gallons does not make sense. The expressionis meaningless.

The expressiontakes hours as input and a number of miles driven as the output. The functionrequires a number of miles as the input. Using(miles driven) as an input value forwhere gallons of gas depends on miles driven, does make sense. The expressionmakes sense, and will yield the number of gallons of gas used,driving a certain number of miles,inhours.[/hidden-answer]

### Are there any situations whereandwould 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.

### Try It

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

A gravitational force is still a force, so makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but 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.

### Using a Table to Evaluate a Composite Function

Using (Figure), evaluateand

1 6 3
2 8 5
3 3 2
4 1 7

To evaluatewe start from the inside with the input value 3. We then evaluate the inside expressionusing the table that defines the function$g\left(3\right)=2.\,$We can then use that result as the input to the functionsois replaced by 2 and we getThen, using the table that defines the functionwe find that

To evaluatewe first evaluate the inside expressionusing the first table:Then, using the table forwe can evaluate

(Figure) shows the composite functionsandas tables.

 3 2 8 3 2

### Try It

Using (Figure), evaluateand

and

#### 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 and 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 theaxis of its graph.
2. Read off the output of the inner function from theaxis of its graph.
3. Locate the inner function output on theaxis of the graph of the outer function.
4. Read the output of the outer function from theaxis of its graph. This is the output of the composite function.

### Using a Graph to Evaluate a Composite Function

Using (Figure), evaluate

We evaluateusing the graph offinding the input of 1 on theaxis and finding the output value of the graph at that input. Here,We use this value as the input to the function

We can then evaluate the composite function by looking to the graph offinding the input of 3 on the axis and reading the output value of the graph at this input. Here,so[/hidden-answer]

#### Analysis

(Figure) shows how we can mark the graphs with arrows to trace the path from the input value to the output value.

### Try It

Using (Figure), evaluate

#### 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 compositionTo do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function likewe 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.

### Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input

Givenandevaluate

Because the inside expression iswe start by evaluating at 1.

Thenso we evaluateat an input of 5.

#### Analysis

It makes no difference what the input variablesandwere called in this problem because we evaluated for specific numerical values.

Givenandevaluate

a. 8; b. 20

### Finding the Domain of a Composite Function

As we discussed previously, the domain of a composite function such asis dependent on the domain ofand the domain ofIt is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such asLet us assume we know the domains of the functionsandseparately. If we write the composite function for an inputaswe can see right away thatmust be a member of the domain ofin order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see thatmust be a member of the domain ofotherwise the second function evaluation incannot be completed, and the expression is still undefined. Thus the domain ofconsists of only those inputs in the domain ofthat produce outputs frombelonging to the domain ofNote that the domain ofcomposed withis the set of allsuch thatis in the domain ofandis in the domain of

### Domain of a Composite Function

The domain of a composite functionis the set of those inputsin the domain offor whichis in the domain of

### How To

Given a function compositiondetermine its domain.

1. Find the domain of
2. Find the domain of
3. Find those inputsin the domain offor whichis in the domain ofThat is, exclude those inputsfrom the domain offor whichis not in the domain ofThe resulting set is the domain of

### Finding the Domain of a Composite Function

Find the domain of

The domain ofconsists of all real numbers exceptsince that input value would cause us to divide by 0. Likewise, the domain ofconsists of all real numbers except 1. So we need to exclude from the domain ofthat value offor which

So the domain ofis the set of all real numbers exceptandThis means that

We can write this in interval notation as

### Finding the Domain of a Composite Function Involving Radicals

Find the domain of

Because we cannot take the square root of a negative number, the domain ofisNow we check the domain of the composite function

For
since the radicand of a square root must be positive. Since square roots are positive, or, which gives a domain of .[/hidden-answer]

#### Analysis

This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain ofcan contain values that are not in the domain ofthough they must be in the domain of

### Try It

Find the domain of

### Decomposing a Composite Function into its Component Functions

In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.

### Decomposing a Function

Writeas the composition of two functions.

We are looking for two functions,andsoTo do this, we look for a function inside a function in the formula forAs one possibility, we might notice that the expressionis the inside of the square root. We could then decompose the function as

We can check our answer by recomposing the functions.

### Try It

Writeas the composition of two functions.

Access these online resources for additional instruction and practice with composite functions.

### Key Equation

 Composite function

### Key Concepts

• We can perform algebraic operations on functions. See (Figure).
• 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 (Figure) and (Figure).
• The order of function composition must be considered when interpreting the meaning of composite functions. See (Figure).
• 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 (Figure).
• A composite function can be evaluated from a graph. See (Figure).
• A composite function can be evaluated from a formula. See (Figure).
• 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 (Figure) and (Figure).
• 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 (Figure).

### Section Exercises

#### Verbal

How does one find the domain of the quotient of two functions,

Find the numbers that make the function in the denominatorequal to zero, and check for any other domain restrictions onandsuch as an even-indexed root or zeros in the denominator.

What is the composition of two functions,

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

How do you find the domain for the composition of two functions,

#### Algebraic

For the following exercises, determine the domain for each function in interval notation.

Givenandfindand

domain:

domain:

domain:

domain:

Givenandfindand

Givenandfindand

domain:

domain:

domain:

domain:

Givenandfindand

Given andfindand

domain:

domain:

domain:

domain:

Givenandfind

For the following exercise, find the indicated function givenand

a. 3; b.c.d.e.

For the following exercises, use each pair of functions to findandSimplify your answers.

For the following exercises, use each set of functions to findSimplify your answers.

$g\left(x\right)=x-6,\,$and

$g\left(x\right)=\frac{1}{x},\,$and

Givenandfind the following:

1. the domain ofin interval notation
2. the domain of

Givenandfind the following:

1. the domain ofin interval notation

a.b.

Given the functionsfind the following:

Given functionsandstate the domain of each of the following functions using interval notation:

a.b.c.

Given functionsandstate the domain of each of the following functions using interval notation.

Forandwrite the domain ofin interval notation.

For the following exercises, find functionsandso the given function can be expressed as

sample:

sample:

sample:

sample:

sample:

sample:

sample:

sample:

#### Graphical

For the following exercises, use the graphs ofshown in (Figure), andshown in (Figure), to evaluate the expressions.

2

5

4

0

For the following exercises, use graphs ofshown in (Figure),shown in (Figure), andshown in (Figure), to evaluate the expressions.

2

1

4

4

#### Numeric

For the following exercises, use the function values forshown in (Figure) to evaluate each expression.

0 7 9
1 6 5
2 5 6
3 8 2
4 4 1
5 0 8
6 2 7
7 1 3
8 9 4
9 3 0

9

4

2

3

For the following exercises, use the function values forshown in (Figure) to evaluate the expressions.

 11 9 7 0 0 5 1 1 3 0 2 1 3

11

0

7

For the following exercises, use each pair of functions to findand

For the following exercises, use the functionsand
to evaluate or find the composite function as indicated.

#### Extensions

For the following exercises, useand

Findand

2

What is the domain of

What is the domain of

Let

1. Find
2. Isfor any functionthe same result as the answer to part (a) for any function? Explain.

For the following exercises, let$f\left(x\right)={x}^{5},\,$and

True or False:

True or False:

For the following exercises, find the composition whenfor alland

;

#### Real-World Applications

The functiongives the number of items that will be demanded when the price isThe production costis the cost of producingitems. To determine the cost of production when the price is \$6, you would do which of the following?

1. Evaluate
2. Evaluate
3. Solve
4. Solve

The functiongives the pain level on a scale of 0 to 10 experienced by a patient withmilligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system afterminutes is modeled byWhich of the following would you do in order to determine when the patient will be at a pain level of 4?

1. Evaluate
2. Evaluate
3. Solve
4. Solve

c

A store offers customers a 30% discount on the priceof selected items. Then, the store takes off an additional 15% at the cash register. Write a price functionthat computes the final price of the item in terms of the original price(Hint: Use function composition to find your answer.)

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according tofind the area of the ripple as a function of time. Find the area of the ripple at

and square inches

A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formulaexpress the area burned as a function of time,(minutes).

Use the function you found in the previous exercise to find the total area burned after 5 minutes.

square units

The radiusin inches, of a spherical balloon is related to the volume,byAir is pumped into the balloon, so the volume afterseconds is given by

1. Find the composite function
2. Find the exact time when the radius reaches 10 inches.

The number of bacteria in a refrigerated food product is given by $3<T<33,\,$where is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by where is the time in hours.

1. Find the composite function
2. Find the time (round to two decimal places) when the bacteria count reaches 6752.

a.
b. 3.38 hours