Linear Equations: Relations and Functions

Linear Equations: Relations and Functions

​Lesson Target

Students will be able to ​Recognize a relation as a set of ordered pairs that relates an input to an output, and a function as relation for which there is exactly one output for each input, using tables by completing a quizizz lecture activity.

Fill in the Blanks

The seventh grade class is planning a field trip. There are two proposals, one to a bowling alley and one to the Omni Theater. To make the best choice, the students have to do some research.

Casey and his friend Max are in charge of researching different bowling alleys to find the best price. They find out that the local bowling alley that is closest to school has a very good offer. This bowling alley charges a flat fee for shoes and then a fee per game.

“I wonder how many games we will have time for,” Casey said to Max.

“I don’t know, but that will impact the cost,” Max responded.

“Let’s figure it out. What is the flat fee for shoes?” Casey asked.

“It is $2.00 and it is $3.00 per game,” Max said.

The two boys took out a piece of paper and began to figure out how much the total would be based on games.

Try and create an equation that would solve for the following problem?

​What is a function?

To solve the last problem, you will need to understand functions. A function is when one variable is impacted by another. In this case, there is a fee for shoes and a fee per game. The total cost per student will depend on the number of games. The cost is a function of the games. Learn all that you can and you will be able to figure out the fees at the end of the lesson.

​How did we get this crazy mess?

​Relations and Functions: Recognizing Relations and Functions​

​In this lesson, we will learn to recognize a relation as a set of ordered pairs that relates an input to an output, and a function as relation for which there is exactly one output for each input, using tables.

​

When we work with relations and functions, we work with the world of relationships. We look at how one factor impacts or effects another factor.

​

What is a relation?

A relation is written as a set of ordered pairs where one value is equal to x and one value is equal to y. What we are really looking at with a relation is the relationship between one factor and another. Let’s think about an example.

​Relations and Functions: Recognizing Relations and Functions​

Draw

Draw out a relation for motorcycles to motorcycle tires on a 2 cycled motorcycle using your own chart?

​Relations and Functions: Recognizing Relations and Functions​

​A motorcycle has an ordered pair of bikes to tires as (1, 2).

This means that for every one motorcycle there are two tires. This is a relation.

Draw

Create a relation between the number of cars a company can sell compared to the number of tires each car has...

​Relations and Functions: Recognizing Relations and Functions​

​A car has an ordered pair of cars to tires as (1,4).

This means that for every one car that is built, there are four tires. This is a relation.

​

​Example:​​

A soup kitchen prepares food for people every day of the month. The supervisor keeps count of the number of people who eat every day. Her data table for the first few days is to the right.

​

She could rewrite this data as a relation, a set of ordered pairs. The first coordinate would be the day of the month and the second coordinate would be the number of visitors.

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​She would show the relation like this {(1, 82), (2, 84), (3, 87), (4, 80), (5, 91), (6, 93), (7, 104), (8, 84), (9, 88)}. Notice that the days of the week form the x value and the number of visitors forms the y value.

​

The braces, {}, indicate that these are all the ordered pairs in the set.​

​There are parts of a relation too. We can have a domain and a range for every relation. The values in the domain and range help us to understand the relation. The domain is made up of the values in first column or the x coordinate in the relation. The range is made up of the second column or the y value of the relation.

​Relations and Functions: Recognizing Relations and Functions​

Draw

The following question has the relation as follows: {(1,4), (2,8), (3,12), (4,16)}. Write out the domain and range of the following relation:

​There are different types of relations too. A relation can be a function or not a function.

​

A function is a relation in which each member of the domain is paired with exactly one member of the range. In other words, a number in the domain cannot have two values for the range.

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In the example to the right, every day of the month has only one number of visitors. Then this relation is a function. When we look at the values in the domain and the range, we can figure out if the relation is a function or not.

​Domain: {1,2,3,4,5,6,7,8,9}

Range: {82,84,87,80,91,93,104,84,88}​

Poll

Is the following image a relation to a function?

• Yes

• No

• I am lost?

Poll

Is the following image a relation to a function?

• Yes

• No

• I do not know I am lost?

In the last image with a relation to a function, the relation is not a function because 12 in the domain is paired with two values in the range.

​We can figure out if a relation is a function by looking at ordered pairs too. Look at this example.

​

Example​

Is this relation a function? {(8, -2), (5, -3), (0, -9), (8, -4)}

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​To figure this out, we look at the values in the domain. The value 8 has two values in the range that are matched with it, so this relations is not a function.

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Some real – life examples are considered functions. Think about the motorcycle example and the number of tires. The number of tires is a function of the motorcycle. We can also say that the amount of carbon dioxide gas that can dissolve in a soda beverage depends on the soda’s temperature. The gas is a function of the temperature.​

Multiple Choice

What is a relation?

• When you marry someone.
• Equations that contain parentheses.
• A set of ordered pairs.
• A set of numbers & variables.

Multiple Choice

What makes a relation a function?

• Each element of the range is paired with exactly one element of the domain.
• Each element of the domain is paired with exactly one element of the range.
• The elements in the domain and range are identical.
• There are no identical elements in the domain and range.

Multiple Choice

What is another name for the set of the second numbers of the ordered pairs in a relation?

• Domain
• Function
• Input
• Range

Multiple Choice

What is another name for the set of the first numbers of the ordered pairs in a relation?

• Domain
• Output
• Range
• Function

Multiple Choice

What is the domain of the given relation:
{(-1, 2),(3, 5),(-5,7)}

• {2, 5, 7}
• {1, 2, 3, 5, -5, 7}
• The domain is not given.
• {-1, 3, -5}

Multiple Choice

What is the range of the given relation?

• {1, 4, 5, 8}
• {-9, -5, 1, 7}
• {-9, -5, 1, 7, 4, 5, 8}
• {4, 5, 8}

Multiple Choice

Is the following relation a function?

• Yes
• No

Poll

I feel confident in moving onto "the function rule" (next part of relations and functions)

• Yes

• No

• I am still confused by input, output, domain, range, and all of it!