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# 7.7: Functions

Difficulty Level: At Grade Created by: CK-12

## Introduction

A Day at the Amusement Park

The month has flown by and in two days Marc and Kara will head home. They have had a terrific time in Boston and while they are happy to be heading home, they are also sad to leave. For a final trip, Grandma and Grandpa decide to go to the amusement park. Everyone is excited.

At the amusement park, different rides take a different number of tickets. Kara and Marc figure out that there are 9 rides that they would like to ride. These are their favorites and any others are just a bonus. Most often the teens will need three tickets for each ride. Each ride is a function of the number of tickets needed. Marc and Kara go up to the ticket booth and come back with 21 tickets each. If they each have 21 tickets, how many rides can they go on for those tickets?

Because each ride is a function of the number of tickets, you will need to know about functions to figure out this problem. Pay attention to this lesson and at the end you will be able to help with the day at the amusement park.

What You Will Learn

In this lesson, you will learn to demonstrate the following skills:

• Identify the domain and range of a simple linear function.
• Evaluate a given function rule using an input-output table.
• Write a function rule from an input-output table.
• Model and solve real-world problems involving patterns of change.

Teaching Time

I. Identify the Domain and Rangle of a Simple Linear Function

We use the word “function” all the time in everyday speech. We say things like “It’s a function of time” or “It’s a function of price.” This is a real life application of a mathematical concept called a function. You will learn how to apply functions to real-world examples, but first let’s look at what a function is and how we can understand it better.

What is a function?

A function is a set of ordered pairs in which the first element in any pair corresponds to exactly one second element.

For example, look at this set of ordered pairs. Notice that braces, {}, are used to surround the set of ordered pairs.

\begin{align*}& \{ (0, 5), (\underline{1}, \underline{6}), (2, 7), (3, 8)\}\\ & \quad \ \ \qquad \uparrow \uparrow\end{align*}

In (1, 6), 1 is the first element and

6 is the second element.

Each of the first elements––0, 1, 2, and 3––corresponds to exactly one second element. So, this set of ordered pairs represents a function.

Let's take a look at another set of ordered pairs.

\begin{align*}& \{ (\underline{2}, 4), (5, 3), (6, 7), (\underline{2}, 8)\}\\ & \ \ \Box \qquad \qquad \qquad \quad \ \Box\end{align*}

The first element, 2, corresponds

to two different second elements––4 and 8.

Since one of the first elements corresponds to two different second elements, the set of ordered pairs above does not represent a function.

We can use these criteria to determine whether or not a series of ordered pairs forms a function.

Now that you know how to identify a function, let’s look at some of the key words associated with functions.

1. Domain
2. Range

The domain of a function is the set of all the first elements in a function. The range is the set of all the second elements in a function.

Let’s look at an example and identify the domain and range of the series of ordered pairs.

Example

The ordered pairs below represent a function

{(0, -10), (2, -8), (4, -6), (6, -4)}

a. Identify the domain of the function.

b. Identify the range of the function.

Consider part \begin{align*}a\end{align*} first.

The domain is the set of all the first elements in the function. These first elements are underlined below.

\begin{align*}\{ (\underline{0}, -10), (\underline{2}, -8), (\underline{4}, -6), (\underline{6}, -4)\}\end{align*}

The domain of this function is {0, 2, 4, 6}.

Next, consider part \begin{align*}b\end{align*}.

The range is the set of all the second elements in the function. These second elements are underlined below.

\begin{align*}\{ (0, \underline{-10}), (2, \underline{-8}), (4, \underline{-6}), (6, \underline{-4})\}\end{align*}

The range of this function is {-10, -8, -6, -4}.

7P. Lesson Exercises

Identify the domain and range of each function.

1. (1, 3) (2, 4) (5, 7) (9, 11)
2. (8, 12) (9, 22) (4, 7) (2, 5)

Take a few minutes to check your work with a friend.

There are different ways to show a function. Ordered pairs are one way to illustrate a function. Let’s look at another way.

II. Evaluate a Given Function Rule Using an Input-Output Table

An input-output table, like the one shown below, can also be used to represent a function. Because of that, we can also call this kind of a table a function table.

Input number \begin{align*}(x)\end{align*} Output number \begin{align*}(y)\end{align*}
0 0
1 3
2 6
3 9

Each pair of numbers in the table is related by the same function rule. That rule is: multiply each input number (\begin{align*}x-\end{align*}value) by 3 to find each output number (\begin{align*}y-\end{align*}value). You can use a rule like this to find other values for this function, too.

Working with function tables and function rules is a lot like being a detective! You have to use the clue of the function rule to complete a table! Patterns are definitely involved in this work.

Now let’s look at how we can use a function rule to complete a table.

Example

The rule for the input-output table below is: add 1.5 to each input number to find its corresponding output number. Use this rule to find the corresponding output numbers for the given input numbers in the table.

Input number \begin{align*}(x)\end{align*} Output number \begin{align*}(y)\end{align*}
0
1
2.5
5
10

To find each missing output number, add 1.5 to each input number. Then write that output number in the table.

Input number \begin{align*}(x)\end{align*} Output number \begin{align*}(y)\end{align*}
0 1.5 \begin{align*}\leftarrow 0+1.5=1.5\end{align*}
1 2.5 \begin{align*}\leftarrow 1.0+1.5=2.5\end{align*}
2.5 4 \begin{align*}\leftarrow 2.5+1.5=4.0\end{align*}
5 6.5 \begin{align*}\leftarrow 5.0+1.5=6.5\end{align*}
10 11.5 \begin{align*}\leftarrow 10.0+1.5=11.5\end{align*}

The table above shows five ordered pairs that match the given function rule. Let’s write the answer in ordered pairs.

The answer is (0, 1.5) (1, 2.5) (2.5, 4) (5, 6.5) (10, 11.5).

Now let’s look at how to create a function table given a rule.

Example

The rule for a function is: multiply each \begin{align*}x-\end{align*}value by 4 and then subtract 2 to find each \begin{align*}y-\end{align*}value. Make a function table that shows three ordered pairs of values for a function that follows this rule.

First, choose three \begin{align*}x-\end{align*}values for the table. You may choose any numbers, but let's select some small numbers that will be easy to work with, such as 1, 2, and 3. Then, to find the \begin{align*}y-\end{align*}values, multiply each of those values by 4 and subtract 2 from that product.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
1 2

\begin{align*}\leftarrow 1 \times 4=4\end{align*}

\begin{align*}4-2=2\end{align*}

2 6

\begin{align*}\leftarrow 2 \times 4=8\end{align*}

\begin{align*}8-2=6\end{align*}

3 10

\begin{align*}\leftarrow 3 \times 4=12\end{align*}

\begin{align*}12-2=10\end{align*}

The table above shows five ordered pairs that match the given function rule.

The answer is (1, 2) (2, 6) (3, 10).

We can also write a function rule in the form of an equation. Just like an equation shows the relationship between values, the function table does too. Let’s look at an example.

Example

The equation \begin{align*}y=\frac{x}{3}+1\end{align*} describes a function. Use this rule to find the missing values in the table below.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 1
3 2
9
8

The table requires you to find the value of \begin{align*}y\end{align*} when \begin{align*}x = 9\end{align*}. To find the missing \begin{align*}y-\end{align*}value, substitute the given \begin{align*}x-\end{align*}value, 9, for \begin{align*}x\end{align*} into the equation. Then solve for \begin{align*}y\end{align*}.

\begin{align*}y &= \frac{x}{3}+1\\ y &= \frac{9}{3}+1\\ y &= 3+1\\ y &= 4\end{align*}

So, when \begin{align*}x = 9, y = 4\end{align*}. This means that (9, 4) is an ordered pair for this function.

The table also requires you to find the value of \begin{align*}x\end{align*} when \begin{align*}y = 8\end{align*}. To find the missing \begin{align*}x-\end{align*}value, substitute the given \begin{align*}y-\end{align*}value, 8, into the equation. Then solve for \begin{align*}x\end{align*} as you would solve any two-step equation.

\begin{align*}y &= \frac{x}{3}+1\\ 8 &= \frac{x}{3}+1\\ 8-1 &= \frac{x}{3}+1-1\\ 7 &= \frac{x}{3}+0\\ 7 &= \frac{x}{3}\\ \\ 7 \times 3 &= \frac{x}{3} \times 3\\ 21 &= \frac{x}{\cancel{3}} \times \frac{\cancel{3}}{1}\\ 21 &= \frac{x}{1}=x\end{align*}

So, when \begin{align*}y = 8, x = 21\end{align*}. This means that (21, 8) is an ordered pair for this function.

The completed table will look like this.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 1
3 2
9 4
21 8

You could say that an equation is another way of writing a function rule.

7Q. Lesson Exercises

1. Complete the table given the rule "add 2". Write the answer in ordered pairs.

Input \begin{align*}(x)\end{align*} Output \begin{align*}(y)\end{align*}
3
5
6

2. Create a function table given the rule \begin{align*}x \ 2 + 3\end{align*} (input times two, plus three). Use three values.

Check your work with a peer. Did you write your answer as a set of ordered pairs?

III. Write a Function Rule From an Input-Output Table

Up until now, you have been given a function rule or an equation. Now it is time to use those detective skills and figure out the rule from a given table.

Remember, thinking like a detective looking for clues will help you write function rules.

One strategy that might be helpful in this task is to use guess and check to figure out the relationship between the values in the domain and the values in the range. Remember the \begin{align*}x\end{align*} values are the domain and the \begin{align*}y\end{align*} values are the range.

Example

Identify the rule for this input-output table. Describe the rule in words.

Input number \begin{align*}(x)\end{align*} Output number \begin{align*}(y)\end{align*}
10 1
20 2
30 3
40 4

Use guess and check to determine how to each pair of values is related.

First, notice that each output number (\begin{align*}y-\end{align*}value) is less than its corresponding input number (\begin{align*}x-\end{align*}value). There are two operations that help numbers become smaller. So, the rule will involve either subtraction or division.

Look for a subtraction rule first.

Consider the ordered pair (10, 1).

\begin{align*}10-1=9\end{align*}, so the rule could be to subtract 9 from each input number to find the corresponding output number. Check to see if that rule works for the other pairs of values in the table.

Consider the ordered pair (20, 2).

\begin{align*}20-1=19\end{align*}, not 2, so that rule does not work all the pairs of values in the table.

Look for a division rule next.

Consider the ordered pair (10, 1) again.

\begin{align*}10 \div 10=1\end{align*}, so the rule could be to divide each input number by 10 to find the corresponding output number. Check to see if that rule works for the other pairs of values in the table.

Consider the ordered pair (20, 2).

\begin{align*}20 \div 10=2\end{align*}, so the rule works for that ordered pair.

Consider the ordered pair (30, 3).

\begin{align*}30 \div 10=3\end{align*}, so the rule works for that ordered pair.

Consider the ordered pair (40, 4).

\begin{align*}40 \div 10=4\end{align*}, so the rule works for that ordered pair.

The rule for this function table is to divide each input number (\begin{align*}x-\end{align*}value) by 10 to find its corresponding output number (\begin{align*}y-\end{align*}value).

In this example, we wrote the function rule in words. We can also use an equation to express a function rule. Think about this past example.

The rule was “Divide each \begin{align*}x\end{align*} value by 10 and that will give you the \begin{align*}y\end{align*} value.”

We could write \begin{align*}\frac{x}{10}=y\end{align*}.

This would have been the same rule expressed as an equation.

Let’s look at another example where we will write an equation to represent a function rule.

Example

Identify the rule for this function table. Describe the rule in words. Then write an equation to represent the relationship between the pairs of values in the table.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
1 5
2 10
3 15
4 20

Use guess and check to determine how each pair of values is related.

For example, notice that each \begin{align*}y-\end{align*}value is greater than its corresponding \begin{align*}x-\end{align*}value. So, the rule will involve either addition or multiplication.

Look for an addition rule first.

Consider the ordered pair (1, 5).

\begin{align*}1+4=5\end{align*}, so the rule could be to add 4 to each \begin{align*}x-\end{align*}value to find the corresponding y-value. Check to see if that rule works for the other pairs of values in the table.

Consider the ordered pair (2, 10).

\begin{align*}2+5=7\end{align*}, not 10, so that rule does not work all the pairs of values in the table.

Look for a multiplication rule next. Each \begin{align*}y-\end{align*}value in the table is 5 more than the previous \begin{align*}y-\end{align*}value, so the rule may involve multiplying by 5.

Consider the ordered pair (1, 5).

\begin{align*}1 \times 5=5\end{align*}, so the rule could be to multiply each \begin{align*}x-\end{align*}value by 5 to find the corresponding \begin{align*}y-\end{align*}value. Check to see if that rule works for the other pairs of values in the table.

Consider the ordered pair (2, 10).

\begin{align*}2 \times 5=10\end{align*}, so the rule works for that ordered pair.

Consider the ordered pair (3, 15).

\begin{align*}3 \times 5=15\end{align*}, so the rule works for that ordered pair.

Consider the ordered pair (4, 20).

\begin{align*}4 \times 5=20\end{align*}, so the rule works for that ordered pair.

The rule for this function table is "multiply each \begin{align*}x-\end{align*}value by 5 to find its corresponding \begin{align*}y-\end{align*}value".

Now that we have a rule in words, let's write an equation to show the same relationship. Remember, to find each \begin{align*}y-\end{align*}value, you must multiply each \begin{align*}x-\end{align*}value by 5. So, the equation would be:

\begin{align*}y=x \cdot 5\end{align*} or \begin{align*}y=5x\end{align*}.

Yes it is. You will also become better at figuring out the rules over time.

One thing to consider is that sometimes, a function rule will involve more than one step. In other words, it will have two operations in it, not just one.

Example

Identify the rule for this function table. Describe the rule in words. Then write an equation to represent that rule.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
1 6
2 11
3 16
4 21

Use guess and check to determine how to each pair of values is related.

For example, notice that each \begin{align*}y-\end{align*}value is greater than its corresponding \begin{align*}x-\end{align*}value. So, the rule will involve either addition or multiplication.

Since each \begin{align*}y-\end{align*}value in the table is 5 more than the previous \begin{align*}y-\end{align*}value, the rule may involve multiplying by 5.

Look for a two-step rule that involves multiplying by 5.

Consider the ordered pair (1, 6).

\begin{align*}1 \times 5=5\end{align*} and \begin{align*}5+1=6\end{align*}, so the rule could be to multiply each input number by 5 and then to add 1 to find each \begin{align*}y-\end{align*}value. Check to see if that rule works for the other pairs of values in the table.

Consider the ordered pair (2, 11).

\begin{align*}2 \times 5=10\end{align*} and \begin{align*}10+1=11\end{align*}, so the rule works for that ordered pair.

Consider the ordered pair (3, 16).

\begin{align*}3 \times 5=15\end{align*} and \begin{align*}15+1=16\end{align*}, so the rule works for that ordered pair.

Consider the ordered pair (4, 21).

\begin{align*}4 \times 5=20\end{align*} and \begin{align*}20+1=21\end{align*}, so the rule works for that ordered pair.

The rule for this function table is to multiply each \begin{align*}x-\end{align*}value by 5 and then add 1 to find its corresponding \begin{align*}y-\end{align*}value.

Now that we have a rule in words, let's write an equation to show the same relationship. Remember, to find each \begin{align*}y-\end{align*}value, we must multiply each \begin{align*}x-\end{align*}value by 5 and then add 1. So, the equation would be

\begin{align*}y=5x+1\end{align*}

7R. Lesson Exercises

Look at each function table and write each rule as an equation.

1.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
5 10
7 14
9 18

2.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
3 5
4 7
6 11

3.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
2 1
4 3
6 5

Take a few minutes to check your answers with a friend. Correct any errors and then continue with the next section.

IV. Model and Solve Real-World Problems Involving Patterns of Change

Functions can also help us describe real-world situations and solve real-world problems.

Example

The number of quarters needed to wash clothes in a washing machine at a laundromat is a function of the number of loads of laundry that need to be washed. This table shows the number of quarters needed to wash 1, 2, and 3 loads of laundry.

Number of Loads \begin{align*}(x)\end{align*} Number of Quarters \begin{align*}(y)\end{align*}
1 4
2 8
3 12
4 ?
5 ?

a. Write an equation to describe the relationship between the pairs of values in the table above.

b. How many quarters are needed to wash 5 loads of laundry?

Consider part \begin{align*}a\end{align*} first.

Use guess and check to determine how to each pair of values is related.

For example, notice that each \begin{align*}y-\end{align*}value is greater than its corresponding \begin{align*}x-\end{align*}value. So, the rule will involve either addition or multiplication.

Since each \begin{align*}y-\end{align*}value in the table is 4 more than its corresponding \begin{align*}y-\end{align*}value, the rule may involve multiplying by 4.

Look for a rule that involves multiplying by 4.

Consider the ordered pair (1, 4).

\begin{align*}1 \times 4=4\end{align*}, so the rule could be to multiply each \begin{align*}x-\end{align*}value (or number of loads) by 4 to find each \begin{align*}y-\end{align*}value (or number of quarters). Check to see if that rule works for the other pairs of values in the table.

Consider the ordered pair (2, 8).

\begin{align*}2 \times 4=8\end{align*}, so the rule works for that ordered pair.

Consider the ordered pair (3, 12).

\begin{align*}3 \times 4=12\end{align*}, so the rule works for that ordered pair.

So, the rule for this function table could be to multiply each \begin{align*}x-\end{align*}value by 4 to find its corresponding \begin{align*}y-\end{align*}value. We could also say that the rule is to multiply the number of loads of laundry by 4 to find the number of quarters needed.

Now that we have our rule written out in words, let’s write an equation that expresses the same thing.

Remember, to find each \begin{align*}y-\end{align*}value, you must multiply each \begin{align*}x-\end{align*}value by 4. So, the equation would be

\begin{align*}y=4x\end{align*}

Next, use the equation you wrote for part \begin{align*}a\end{align*} to solve part \begin{align*}b\end{align*}.

Part \begin{align*}b\end{align*} asks how many quarters are needed to wash 5 loads of laundry. Substitute 5 for \begin{align*}x\end{align*} in the equation.

\begin{align*}y &= 4x\\ y &= 4 \cdot 5\\ y &= 20\end{align*}

20 quarters would be needed to wash 5 loads of laundry.

Now we can go back to the problem in the introduction and help with the amusement park tickets.

## Real Life Example Completed

A Day at the Amusement Park

Here is the original problem once again. Reread it and underline any important information.

The month has flown by and in two days Marc and Kara will head home. They have had a terrific time in Boston and while they are happy to be heading home, they are also sad to leave. For a final trip, Grandma and Grandpa decide to go to the amusement park. Everyone is excited.

At the amusement park, different rides take a different number of tickets. Kara and Marc figure out that there are 9 rides that they would like to ride. These are their favorites and any others are just a bonus. Most often the teens will need three tickets for each ride. Each ride is a function of the number of tickets needed. Marc and Kara go up to the ticket booth and come back with 21 tickets each. If they each have 21 tickets, how many rides can they go on for those tickets?

The first thing to note is that each ride is a function of the number of tickets needed. We can call the ride \begin{align*}x\end{align*} and the ticket number \begin{align*}y\end{align*}.

You know from this lesson that one of the best ways to work with a function is through a function table. Let’s build one now.

\begin{align*}x\end{align*} Rides \begin{align*}y\end{align*} Tickets

We can say that 1 ride is equal to 3 tickets. Let’s choose some other ride numbers and work our way up to 21 tickets.

\begin{align*}x\end{align*} Rides \begin{align*}y\end{align*} Tickets
1 3
2 6
3 9
21

Look at the pattern. We can write an equation to show this pattern.

\begin{align*}y=3x\end{align*}

We want to figure out how many rides the teens can go on for 21 tickets. 21 is the \begin{align*}y\end{align*} in the equation. We can use what we have learned about solving equations to solve for \begin{align*}x\end{align*}. This will give us the number of rides they can go on for 21 tickets.

\begin{align*}21 &= 3x\\ 7 &= x\end{align*}

\begin{align*}x\end{align*} Rides \begin{align*}y\end{align*} Tickets
1 3
2 6
3 9
7 21

The teens can go on 7 rides each for the 21 tickets. To ride all nine rides, they will need six more tickets each.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Function
A set of ordered pairs in which one element corresponds to exactly one other element. Functions might be expressed as a set of ordered pairs or in a table.
Domain
the \begin{align*}x\end{align*} value of an ordered pair or the \begin{align*}x\end{align*} values in a set of ordered pairs.
Range
the \begin{align*}y\end{align*} value of an ordered pair or the \begin{align*}y\end{align*} values in a set of ordered pairs.
Input-Output Table
a way of showing a function using a table where the \begin{align*}x\end{align*} value is shown to cause the \begin{align*}y\end{align*} value through a function rule.
Function Table
another name for an input-output table.
Function Rule
a written equation that shows how the domain and range of a function are related through operations.

## Time to Practice

Directions: Identify whether or not each series of ordered pairs forms a function.

1. (1, 3)(2, 6)(2, 5) (3, 7)

2. (2, 5) (3, 6) (4, 7) (5, 8)

3. (6, 1) (7, 2) (8, 3)

4. (5, 2) (5, 3) (5, 4) (5, 5)

5. (81, 19)(75, 18) (76, 18) (77, 19)

Directions: Identify the domain in numbers 1 – 5.

6.

7.

8.

9.

10.

Directions: Identify the range in numbers 1 – 5.

11.

12.

13.

14.

15.

Directions: Use the given rule or equation to complete the table.

16. The rule for the input-output below table is: multiply each input number by 7 and then add 2 to find each output number. Use this rule to find the corresponding output numbers for the given input numbers in the table. Fill in the table with those numbers.

Input number \begin{align*}(x)\end{align*} Output number \begin{align*}(y)\end{align*}
0
1
2
3
4

17. The rule for this function table is: subtract 6 from each \begin{align*}x-\end{align*}value to find each \begin{align*}y-\end{align*}value. Use this rule to find the missing numbers in the table. Fill in the table with those numbers.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0
6
7
16

18. The equation \begin{align*}y=\frac{x}{2}-1\end{align*} describes a function. Use this rule to find the missing values in the table below.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
2 0
4 1
8
6

Directions: Determine the rule for each table. Use that rule to answer the questions.

19. Identify the rule for this input-output table. Describe the rule in words.

Input number \begin{align*}(x)\end{align*} Output number \begin{align*}(y)\end{align*}
1 3
2 4
3 5
4 6

20. Identify the rule for this function table.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
1 2
2 4
3 6
4 8

a. Describe the rule in words.

b. Write an equation to represent the rule.

21. Identify the rule for this function table.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
1 3
2 5
3 7
4 9

a. Describe the rule in words.

b. Write an equation to represent the rule.

Directions: Solve each problem.

22. The total number of cupcakes Shakir can bake, \begin{align*}y\end{align*}, is a function of the number of batches of cupcake batter he makes, \begin{align*}x\end{align*}. This table shows the total number of cupcakes Shakir will bake if he makes 1, 2 or 3 batches of cupcake batter.

Number of Batches of Batter \begin{align*}(x)\end{align*} Total Number of Cupcakes Baked \begin{align*}(y)\end{align*}
1 12
2 24
3 36
4 ?
5 ?

a. Write a rule in words to describe the relationship between the pairs of values in the table above.

b. Write an equation to describe the relationship between the pairs of values in the table above.

c. What is the total number of cupcakes that Shakir will bake if he makes 5 batches of batter?

23. For a concert in the auditorium, Ms. Walsh set up 10 chairs on the stage for the performers and 20 chairs in each row for the audience. This equation shows the relationship between \begin{align*}r\end{align*}, the number of rows of chairs Ms. Walsh set up and \begin{align*}t\end{align*}, the total number of chairs set up in the auditorium:

\begin{align*}t=10+20r\end{align*}.

a. Create a table to show the total number of chairs that would be set up in the auditorium if Ms. Walsh set up 0, 1, 2, 3, or 4 rows of chairs.

b. Identify the domain for the ordered pairs in the table.

c. Explain why the domain shown in the table is a reasonable domain for this function and why including a number less than 0 would not be reasonable.

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