9.1: Relations and Functions
Introduction
The Bowling Field Trip
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.
To solve this 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.
What You Will Learn
In this lesson, you will learn the following skills.
- 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.
- Recognize the domain of a function as the set of all possible input values, the range of a function as the set of all possible output values, and the function rule as an equation assigning each value in the domain to exactly one value in the range.
- Evaluate a function rule by finding outputs for given inputs.
- Write a function rule for given inputs and outputs.
- Write and evaluate function rules to model real – world relations.
Teaching Time
I. Recognizing Relations and Functions
Many numbers have precise and predictable relationships—the number of motorcycles and the number of tires, the numbers of hours you work and the money you get paid, the number of years you go to college and your lifetime earnings.
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 \begin{align*}x\end{align*} and one value is equal to \begin{align*}y\end{align*}. What we are really looking at with a relation is the relationship between one factor and another. Let’s think about an example.
Example
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.
Let’s look at another example.
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 below.
Day of Month | # of Visitors |
---|---|
1 | 82 |
2 | 84 |
3 | 87 |
4 | 80 |
5 | 91 |
6 | 93 |
7 | 104 |
8 | 84 |
9 | 88 |
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.
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 \begin{align*}x\end{align*} value and the number of visitors forms the \begin{align*}y\end{align*} 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 \begin{align*}x\end{align*} coordinate in the relation. The range is made up of the second column or the \begin{align*}y\end{align*} value of the 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. In the example above, 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.
Write the definitions for relation, domain, range and function in your notebook.
Example
Is this relation a function?
\begin{align*}& 5 \qquad 10\\ & 4 \qquad 11\\ & 3 \qquad 9\\ & 6 \qquad 13\\ & 8 \qquad 4\\ & 7 \qquad 3\end{align*}
To figure this out, we have to compare the values from the first column with the values in the second column. For each value of the domain, there is exactly one value in the range. In other words, there aren’t any values that repeat.
Therefore, this relation is a function.
Example
Is this relation a function?
\begin{align*}& 5 \qquad 18\\ & 6 \qquad 19\\ & 12 \quad \ \ 24\\ & 12 \quad \ \ 13\end{align*}
This 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)}
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.
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.
Let’s continue to look at and understand functions.
II. Recognize the domain of a function as the set of all possible input values, the range of a function as the set of all possible output values, and the function rule as an equation assigning each value in the domain to exactly one value in the range
One of the great things about functions is that they can be applied to all kinds of situations. Just remember that in order for a relation to be a function, that the values of the domain need to be assigned to only one value of the range. One way of thinking about functions is through the use of function tables. A function table is an input/output table where the input is the domain and the output is the range. Look at this table below.
Input | Output |
---|---|
3 | 6 |
4 | 8 |
5 | 11 |
6 | 12 |
Here is function where we have an input and an output. The input is the domain and the output is the range. If we were going to write this function as ordered pairs, we would use the input for the \begin{align*}x\end{align*} value and the output as the \begin{align*}y\end{align*} value.
Let’s write out this relation: (3, 6) (4, 8) (5, 10) (6, 12). There is a relationship between the values of the domain and the values of the range. We can say that the range was created when some operation or operations was completed with the domain value.
The output is a result of an operation to the input!
What happened to the input to equal the output?
If you think about this, you will see that the \begin{align*}x\end{align*} value was multiplied by 2 or doubled to equal the \begin{align*}y\end{align*} value. We can write this as an equation.
\begin{align*}y = 2x \end{align*}
This says that the value of \begin{align*}y\end{align*} is created whenever the \begin{align*}x\end{align*} value is multiplied by 2.
This is called a Function Rule. It can be written in words or in the form of an equation. The function rule tells you what operation or operations to perform with the input to get the output.
III. Evaluate a Function Rule by Finding Outputs for Given Inputs
Now that you understand what a function rule is, we can use function rules to complete the outputs of different tables. Let’s look at an example.
Example
Use the function rule \begin{align*}3x\end{align*} to evaluate the given inputs and complete each output.
Input | Output |
---|---|
2 | |
3 | |
4 | |
5 |
Now not all inputs will be this simple, but it will give you some practice applying a function rule. We know that the function rule is \begin{align*}3x\end{align*}, so we can take each value from the input column and multiply it by 3. This will give us the correct value for the output table.
Input | Output |
---|---|
2 | 6 |
3 | 9 |
4 | 12 |
5 | 15 |
You can see that the function rule was applied to each input value and the resulting output values complete the table. Let’s look at another example.
Example
Use the function rule \begin{align*}2x + 1\end{align*} to evaluate each input value. Complete the given table.
Input | Output |
---|---|
\begin{align*}-2\end{align*} | |
-1 | |
0 | |
1 | |
2 |
This table has negative and positive input values, but we will follow the same procedure. Simply substitute each \begin{align*}x\end{align*} value into the function rule and evaluate for the output value.
\begin{align*}2(-2)+1 &=-4+1=-3\\ 2(-1)+1 &=-2+1=-1\\ 2(0)+1 &=0+1=1\\ 2(1)+1 &=2+1=3\\ 2(2)+1 &=4+1=5\end{align*}
Now we can substitute those values into the output column of our function.
Input | Output |
---|---|
\begin{align*}-2\end{align*} | -3 |
-1 | -1 |
0 | 0 |
1 | 1 |
2 | 3 |
This is the answer and our work is complete.
Remember that we call this work evaluating a function rule.
IV. Write a Function Rule for Given Input and Outputs
In the last section, you used function rules to evaluate function tables and find outputs. We can also use input/output tables to help us to write function rules. When we look at the input and decipher what happened to it to create the output, then we can write a function rule based on our discoveries. This is like being a detective! You will have to use what you have learned and look for clues.
Example
\begin{align*}x\end{align*} | \begin{align*}f(x)\end{align*} |
---|---|
0 | 5 |
3 | 8 |
6 | 11 |
9 | 14 |
The first thing to notice is that the words input and output have been replaced by the \begin{align*}x\end{align*} and the \begin{align*}f(x)\end{align*}. This means that we are using function notation to say that \begin{align*}x\end{align*} is the input and that the output is a function of \begin{align*}x\end{align*}. That is what the \begin{align*}f(x)\end{align*} means. Function notation will affect how we write our rule too, but let’s look at that in a minute.
That is exactly what you need to do. In looking at this pattern, you can see that the \begin{align*}x\end{align*} value is increased in each step of the table. Each \begin{align*}x\end{align*} value is increased by 5. This means that we can write the following rule for our function.
\begin{align*}f(x) = x + 5\end{align*}
This is our answer.
Write down an example of function notation and that you need to look for a pattern when figuring out function rules. Put this information in your notebook.
Example
Use function notation to write a function rule for the given table.
\begin{align*}x\end{align*} | \begin{align*}f(x)\end{align*} |
---|---|
12 | 6 |
9 | 4.5 |
7 | 3.5 |
4 | 2 |
What pattern do you notice? Each value of the input has been divided in half. We can write this function rule in two ways.
\begin{align*}f(x)&= \frac{1}{2}x \\ or \ f(x)&= \frac{x}{2}\end{align*}
Both of these will work as a correct answer.
V. Write and Evaluate Function Rules to Model Real – World Relations
Functions like these are seen in countless real-life situations. You have used many yourself without even thinking about it. Let’s look at how functions apply to real world situations.
Example
You are going to order a bunch of churros from a vender for your family. Each one costs $1.50. How much will they cost if you buy 6 or 8 or 10? What’s the function rule?
First, we can take the given information to write the rule.
\begin{align*}p(c)=1.50c \end{align*} where \begin{align*}p\end{align*} is total price and \begin{align*}c\end{align*} is the number of churros.
Next, we can we can substitute different values into the function rule to figure out the cost for 6, 8 or 10 churros.
\begin{align*} & p(c) =1.50c && p(c)=1.50c && p(c) =1.50c\\ & p(6) =1.50 \cdot 6 && p(8)=1.50 \cdot 8 && p(10)=1.50 \cdot 10\\ & p(6) =9 && p(8) =12 && p(10)=15\\ & \$ 9 \text{ for } 6 \text{ churrors} && \$ 12 \text{ for } 8 \text{ churros} && \$ 15 \text{ for } 10 \text{ churros}\end{align*}
Based on the number of churros, we can figure out the differences in cost. This is our answer.
Now let’s go back to the problem from the introduction and look at applying what we have learned about relations and functions to our work.
Real-Life Example Completed
The Bowling Field Trip
Here is the original problem once again. To finish this problem, write an equation that represents the scenario. Then create a table of values to show the varying costs.
The seventh grade class is planning a field trip to a bowling alley. 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.
Remember, there are two parts to your problem.
Solution to Real – Life Example
The first thing to do is to write an equation that represents the given information. We know that each game is $3.00 and the flat rate for shoes is $2.00. The varying value is the cost and that is impacted by the number of games played. The number of games played is our variable.
\begin{align*}C(g) = 3g + 2\end{align*}
This equation means that \begin{align*}c\end{align*} the cost is a function of the number of games plus the $2 shoe fee.
Games Played | Cost ($) |
---|---|
2 | 8 |
4 | 14 |
5 | 17 |
6 | 20 |
The data show that when the number of games increases by 2, the cost increases by $6. Based on the number of games played, the cost could be anywhere from $8 to $20.00 although it is unlikely that any student would have time for 6 games.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Relation
- a set of ordered pairs.
- Domain
- the \begin{align*}x\end{align*} value in a table or function.
- Range
- the \begin{align*}y\end{align*} value in a table or function.
- Function
- Each value in the domain is connected to only one value in the range.
- Function Rule
- the operation or operations performed on the input value which then equals the output value.
- Input
- the \begin{align*}x\end{align*} value or the domain of a function.
- Output
- the \begin{align*}y\end{align*} value or the range of a function.
Time to Practice
Directions: Are the relations functions? If not, explain why.
- {(4, 7), (8, 11), (4, 9), (8, 13)}
- {(-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 1)}
- {(6, 25), (12, 35), (18, 45), (24, 55)}
- {(2, 4), (3, 5) (2, 6), (7, 9)}
- 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.
Directions: Answer each question about functions.
- A pastry chef needs to purchase enough dough for her cookies. She buys one pound of dough for every twenty cookies she is going to make. She uses the function \begin{align*}d(c)=\frac{c}{20}\end{align*} where \begin{align*}c\end{align*} is the number of cookies and \begin{align*}d\end{align*} is the pounds of dough she should buy. Identify which variable is the domain and which is the range.
- Evaluate the function \begin{align*}f(x)=2x+7\end{align*} when the domain is {-3, -1, 1, 3}.
- Evaluate the function \begin{align*}f(x)=\frac{2}{5}x-6\end{align*} when the domain is {-10, -5, 0, 5, 10}.
- Evaluate the function \begin{align*}f(x)=3x-1\end{align*} when the domain is {5, 6, 7, 8, 9}.
- Evaluate the function \begin{align*}f(x)=x-9\end{align*} when the domain is {1, 2, 3, 4, 5}.
- You can convert Celsius degrees to Fahrenheit degrees with the function \begin{align*}F=\frac{9}{5}C+32\end{align*}. Convert \begin{align*}7^{\circ} C, 14^{\circ} C,\end{align*} and \begin{align*}25^{\circ}C\end{align*} to degrees Fahrenheit.
Directions: Write function rules.
- Write a function rule for the following data.
\begin{align*}x\end{align*} | \begin{align*}f(x)\end{align*} |
---|---|
9 | 27 |
11 | 33 |
15 | 45 |
16 | 48 |
- Write a function rule for the following data:
\begin{align*}x\end{align*} | \begin{align*}f(x)\end{align*} |
---|---|
\begin{align*}-2\end{align*} | -6 |
0 | 2 |
2 | 10 |
4 | 18 |
- Write a function rule for the following data:
\begin{align*}x\end{align*} | \begin{align*}f(x)\end{align*} |
---|---|
0 | 0 |
1 | 2 |
4 | 8 |
5 | 10 |
- Write a function rule for the following table.
\begin{align*}x\end{align*} | \begin{align*}f(x)\end{align*} |
---|---|
1 | 0 |
2 | 2 |
4 | 6 |
8 | 14 |
- Write a function rule for the following table.
\begin{align*}x\end{align*} | \begin{align*}f(x)\end{align*} |
---|---|
2 | 1 |
4 | 2 |
8 | 4 |
10 | 5 |
18 | 9 |
- Write a function rule for each table.
\begin{align*}x\end{align*} | \begin{align*}f(x)\end{align*} |
---|---|
6 | 3 |
9 | 3 |
15 | 5 |
21 | 7 |
30 | 10 |
Directions: Solve the following problem.
- Sandwich cost $3.45 each. Write a function rule for the cost, \begin{align*}c\end{align*}, for a number of sandwiches, \begin{align*}s\end{align*}.
- Now, find the cost of 3, 6, and 9 sandwiches.
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