7.16: Input-Output Tables for Function Rules
Remember the car wash from an earlier Concept? Well, here is the problem once again.
Kara and Marc decided to spend one morning helping out at a car wash to benefit a club that their grandparents have participated in for year.
The car wash was a busy place. At the beginning there weren’t any cars, but between 9 am and 10 am the class washed 5 cars. From 10 to 11, the class washed 10 cars, from 11 to 12 the class washed 15 cars and from 12 – 1 the class washed 20 cars.
Marc kept track of the cars washed each hour.
Hour 1 - 5 cars
Hour 2 - 10 cars
Hour 3 - 15 cars
Hour 4 - 20 cars
The number of cars washed is connected to the number of hours worked. One is a function of the other.
Look at this list and write a function rule.
Then figure out how many cars would be washed in the fifth hour of the car wash.
This Concept will show you how to accomplish this task.
Guidance
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.
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.
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 one.
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.
Use what you have learned to complete the following examples.
Example A
Complete the table given the rule add 2.
Input \begin{align*}(x)\end{align*} | Output \begin{align*}(y)\end{align*} |
---|---|
3 | |
5 | |
6 |
Solution: The answers are 5, 7 and 8.
Example B
Write the rule for the table.
Solution:\begin{align*}y = x + 2\end{align*}
Example C
Create a function table given the rule \begin{align*}y = x \div 2 + 3\end{align*}. Use three values.
Solution: Look at the data below.
Input \begin{align*}(x)\end{align*} | Output \begin{align*}(y)\end{align*} |
---|---|
2 | 4 |
4 | 5 |
6 | 5 |
Here is the original problem once again.
Kara and Marc decided to spend one morning helping out at a car wash to benefit a club that their grandparents have participated in for year.
The car wash was a busy place. At the beginning there weren’t any cars, but between 9 am and 10 am the class washed 5 cars. From 10 to 11, the class washed 10 cars, from 11 to 12 the class washed 15 cars and from 12 – 1 the class washed 20 cars.
Marc kept track of the cars washed each hour.
Hour 1 - 5 cars
Hour 2 - 10 cars
Hour 3 - 15 cars
Hour 4 - 20 cars
The number of cars washed is connected to the number of hours worked. One is a function of the other.
Look at this list and write a function rule.
Then figure out how many cars would be washed in the fifth hour of the car wash.
To write the rule, notice that each hour the number of cars increased by 5.
We can use multiplication to show this rule.
\begin{align*}y = 5x\end{align*}
This rule works for all of the values.
Then we can evaluate using the fifth hour.
\begin{align*}5(5) = 25\end{align*}
Given this rule, 25 cars will be washed in hour five.
Vocabulary
Here are the vocabulary words in this Concept.
- Function
- A set of ordered pairs in which one element corresponds to exactly one other element. Functions can 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.
Guided Practice
Here is one for you to try on your own.
At the amusement park, Taylor noticed that there seemed to be a pattern for people who won the dart throwing game. She was so curious that she watched people play the game for a few hours. When 12 people played, there were only 6 winners. When ten people played, there were five winners.
This is a table to represent the data that Taylor collected.
Input | Output |
---|---|
12 | 6 |
10 | 5 |
8 | 4 |
6 | 3 |
4 | 2 |
Can you write a rule for this data?
Answer
We can accomplish this task by looking at what happened to the \begin{align*}x\end{align*} value to get the \begin{align*}y\end{align*} value.
Notice that the \begin{align*}x\end{align*} value was divided by 2.
\begin{align*}\frac{x}{2} = y\end{align*}
This is our rule.
Video Review
Here is a video for review.
- This Khan Academy video is an introduction to functions.
Practice
Directions: Use the given rule or equation to complete the table.
1. 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 |
2. 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 |
3. 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: Evaluate each function rule.
4. \begin{align*}2x\end{align*}
Input | Output |
---|---|
1 | 3 |
2 | 5 |
3 | 7 |
5. \begin{align*}3x-1\end{align*}
Input | Output |
---|---|
1 | 2 |
2 | 5 |
3 | 8 |
4 | 11 |
6. \begin{align*}2x+1\end{align*}
Input | Output |
---|---|
1 | 3 |
2 | 4 |
3 | 6 |
5 | 10 |
7. \begin{align*}4x\end{align*}
Input | Output |
---|---|
0 | 0 |
1 | 4 |
2 | 8 |
3 | 12 |
8. \begin{align*}6x-3\end{align*}
Input | Output |
---|---|
1 | 3 |
2 | 9 |
3 | 15 |
9. \begin{align*}2x\end{align*}
Input | Output |
---|---|
0 | 0 |
1 | 2 |
2 | 4 |
3 | 6 |
10. \begin{align*}3x-3\end{align*}
Input | Output |
---|---|
1 | 0 |
2 | 3 |
4 | 9 |
5 | 12 |
Directions: Create a table for each rule.
11. \begin{align*}7x\end{align*}
12. \begin{align*}3x+1\end{align*}
13. \begin{align*}5x - 3\end{align*}
14. \begin{align*}4x+3\end{align*}
15. \begin{align*}4x- 5\end{align*}
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Here you'll learn to evaluate a given function rule using an input-output table.