Have you ever been bowling? Take a look at this field trip dilemma.
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 Concept.
Guidance
Do you remember how to identify a function, a relation, a range and a domain?
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. When we look at the values in the domain and the range, we can figure out if the relation is a function or not.
A set of ordered pairs is a relation.
The values in the domain and range help us to understand a 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.
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.
Now that you understand how to identify a function rule, let's look at applying this information.
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.
Look at each list of values. Write the output for each list by using \begin{align*}2x-2\end{align*} as the function rule.
Example A
\begin{align*}4, 5, 7, 9\end{align*}
Solution: \begin{align*}6, 8, 12, 16\end{align*}
Example B
\begin{align*}-3, 4, -5, 7\end{align*}
Solution: \begin{align*}-8, 6, -12, 12\end{align*}
Example C
\begin{align*}-1, -9, 11, 12\end{align*}
Solution: \begin{align*}-4, -20, 20, 22\end{align*}
Now let's go back to the dilemma at the beginning of the Concept.
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
- 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.
Guided Practice
Here is one for you to try on your own.
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*} | |
\begin{align*}-1\end{align*} | |
\begin{align*}{\color{white}-}0\end{align*} | |
\begin{align*}{\color{white}-}1\end{align*} | |
\begin{align*}{\color{white}-}2\end{align*} |
Solution
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*} | \begin{align*}\boldsymbol{-3}\end{align*} |
\begin{align*}-1\end{align*} | \begin{align*}\boldsymbol{-1}\end{align*} |
\begin{align*}{\color{white}-}0\end{align*} | \begin{align*}{\color{white}-}\boldsymbol{0}\end{align*} |
\begin{align*}{\color{white}-}1\end{align*} | \begin{align*}{\color{white}-}\boldsymbol{1}\end{align*} |
\begin{align*}{\color{white}-}2\end{align*} | \begin{align*}{\color{white}-}\boldsymbol{3}\end{align*} |
This is the answer and our work is complete.
Video Review
Practice
Directions: For numbers 1 - 5, find each output if the function rule is \begin{align*}3x+2\end{align*}
Problem Number | Input | Output |
---|---|---|
1. | 3 | |
2. | 5 | |
3. | 6 | |
4. | 9 | |
5. | 11 |
Directions: For numbers 6 - 8, find each output if the function rule is \begin{align*}4x\end{align*}
Problem Number | Input | Output |
---|---|---|
6. | -3 | |
7. | -4 | |
8. | 0 | |
9. | 1 | |
10. | 2 |
Directions: For numbers 11 - 15, find each output if the function rule is \begin{align*}-3x\end{align*}
Problem Number | Input | Output |
---|---|---|
11. | 4 | |
12. | 5 | |
13. | 7 | |
14. | 9 | |
15. | 10 |
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}.