Tammy has opened an ice cream stand to earn money for her college tuition. She records the number of cones she sells each day in a table like the one shown below.

Days in operation |
Number of cones sold |

1 | 48 |

2 | 56 |

3 | 77 |

4 | 79 |

5 | 82 |

6 | 98 |

7 | 105 |

Tammy is very pleased with the numbers of cones sold for her first week in operation. Now she wants to analyze the results shown in the table. What information can Tammy record from her table?

In this concept, you will learn to recognize functions.

### Functions

Many things in the real world consist of a relationship between two things. The number of hours a person works and the amount of money earned, the number of gallons of gas bought and the distance a person can drive and the names of the students in a class and their heights are just a few examples of relationships between two things.

The names of the students in a class paired with their heights is an example of a relation. These two things can be paired as (name, height) or as (height, name). The name and the height are written as an ordered pair which simply means that one comes first and the other comes second. When a relationship is expressed as an ordered pair it is called a **relation**. A set of ordered pairs is a **relation**.

If the relationship between the names of the students and their heights is written as (name, height) with the name as the first thing in the ordered pair, then the names are the set of all staring points. This set of starting points is called the **domain** of the relation. The heights are the second thing in the ordered pair and are the set of all ending points. This set of ending points is called the **range** of the relation.

If the variable ‘

’ represents the names of the students and the variable ‘ represents the heights of the students, then the ordered pair (name, height) can be written as . The set of values for ‘ ’ will be the domain and the set of values for ‘ ’ will be the range.Look at the following set of ordered pairs:

\begin{align*}(1,4),(2,6),(2,8),(3,10),(4,12)\end{align*}

The values are written as ordered pairs

such that the domain includes the values and the range includes the values . Notice that the -value of 2 is paired with two different -values. A relation can have a member of the domain paired with more than one member from the range.Look at this set of ordered pairs:

\begin{align*}(1,3),(2,5),(3,7),(4,9),(5,11)\end{align*}

The values are written as ordered pairs **function**.

### Examples

#### Example 1

Earlier, you were given a problem about Tammy and her ice cream cones. She wants to record information from the table.

Tammy can record the domain and range from the table and determine if the information represents a relation or a function.

First, make a list of ordered pairs to represent the information in the table.

\begin{align*}\{(1,48),(2,56),(3,77),(4,79),(5,82),(6,98),(7,105) \}\end{align*}

Next, list the members of the domain. Remember to list each

-value only once.\begin{align*}(1,2,3,4,5,6,7)\end{align*}

Next, list the members of the domain. Remember to list each

-value only once.\begin{align*}(48,56,77,79,82,98,105)\end{align*}

Next, look at the ordered pairs to see if any \begin{align*}y\end{align*}-value.

-value is paired with more than oneEvery

-value from the domain is paired with only one -values from the range.Then, state if the ordered pairs represent a relation or a function.

A function.

The answer is a function.

#### Example 2

Determine if the following represents a function or a relation.

First, follow each arrow from its value in the domain to its value in the range.

Next, write a set of ordered pairs to represent the arrows. Remember the domain represents the

-values and the range represents the -values.

Next, check to see if each

-value is paired with one -value.Then, state if the ordered pairs represent a relation or a function.

A function.

The answer is a function.

#### Example 3

Do the following ordered pairs represent a relation or a function?

\begin{align*}\{ (0,5),(1,6),(2,7),(1,8),(3,9),(2,10) \}\end{align*}

First, list the members of the domain. Remember to list each

-value only once.\begin{align*}(0,1,2,3)\end{align*}

Next, list the members of the domain. Remember to list each

-value only once.\begin{align*}(5,6,7,8,9,10)\end{align*}

Next, look at the ordered pairs to see if any

-value is paired with more than one -value.and . The -value of 1 is paired with two different -values 6 and 8.

and . The -value of 2 is paired with two different -values 7 and 10.

Then, state if the ordered pairs represent a relation or a function.

A relation.

The answer is a relation.

#### Example 4

Do the following ordered pairs represent a relation or a function?

\begin{align*}\{ (0,5),(1,5),(2,5),(3,5) \}\end{align*}

First, list the members of the domain. Remember to list each

-value only once.\begin{align*}(0,1,2,3)\end{align*}

Next, list the members of the domain. Remember to list each

-value only once.

Next, look at the ordered pairs to see if any \begin{align*}x\end{align*}-value is paired with more than one -value.

Each

-value is paired with only one -value.Then, state if the ordered pairs represent a relation or a function.

A function.

The answer is a function.

#### Example 5

Do the following ordered pairs represent a relation or a function?

\begin{align*}\{(8, -2), (5, -3), (0, -9), (8, -4)\}\end{align*}

First, list the members of the domain. Remember to list each

-value only once.\begin{align*}(0,5,8)\end{align*}

Next, list the members of the domain. Remember to list each

-value only once.\begin{align*}(-9,-4,-3,-2)\end{align*}

Next, look at the ordered pairs to see if any

-value is paired with more than one -value.The

-value of 8 is paired with two -values -2 and -4.Then, state if the ordered pairs represent a relation or a function.

A relation.

The answer is a relation.

### Review

Are the relations functions? Write function if it is a function and not a function if it is not a function.

- The amount of bananas you buy at a store for $.85 per pound.
- The amount of carrots that you buy at a store for $.29 per pound.
- The steady price increase of a bus ticket over time.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.1.