7.16: InputOutput 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 inputoutput 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*}
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 inputoutput 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*}
First, choose three \begin{align*}x\end{align*}
\begin{align*}x\end{align*} 
\begin{align*}y\end{align*} 


1  2 
\begin{align*}\leftarrow 1 \times 4=4\end{align*}
\begin{align*}42=2\end{align*} 
2  6 
\begin{align*}\leftarrow 2 \times 4=8\end{align*}
\begin{align*}82=6\end{align*} 
3  10 
\begin{align*}\leftarrow 3 \times 4=12\end{align*}
\begin{align*}122=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*}
\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*}
\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*}
The table also requires you to find the value of \begin{align*}x\end{align*}
\begin{align*}y &= \frac{x}{3}+1\\ 8 &= \frac{x}{3}+1\\ 81 &= \frac{x}{3}+11\\ 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.
 InputOutput 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 inputoutput 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 inputoutput 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*}3x1\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*}6x3\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*}3x3\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 inputoutput table.