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# Function Rules for Input-Output Tables

## Use data in tables to write rules that have one output for any input.

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Function Rules for Input-Output Tables
Credit: Michael Gil
Source: https://www.flickr.com/photos/msvg/8565777646/in/photolist-5KM6MF-jjtyrn-ntzPnD-qbuJ6P-c9GG3S-o54gg6-7s6dDD-7sabuf-7sabTC-7sabCj-7s6e84-cCkD7m-e3VRgu-dPB5Ky-dPB5qG-9iiwBj-5Fn6kt-diqbv5-8FFMgM-7aHi1S-5xELcQ-nBspc-9Z3Sxo
License: CC BY-NC 3.0

Geri is getting ready to go through the automatic car wash with her mom. Geri’s mom gives her several dollar bills and tells her to use it to get quarters for the car wash. Geri puts a dollar in the change machine and the machine gives her 4 quarters. She puts another dollar in the machine and it gives her 4 more quarters. Geri continues until she has 26 quarters. As she walks back to the car, she begins to think about what she put into the change machine and what came out.

This is a table to represent the change machine’s input and output.

 Input (dollars) Output (quarters) 6 24 5 20 4 16 3 12

What rule could Geri write to represent what happened to the input to equal the output?

In this concept, you will learn to evaluate and write function rules for an input-output table.

### Writing Function Rules for Input-Output Tables

A function is when one variable or term depends on another according to a rule. There is a special relationship between the two variables of the function where each value in the input applies to only one value in the output. These rules are called function rules, because they explain how the function operates. The function rule is the same thing as the expression. Here are some hints for writing function rules:

1. Decipher the pattern of the function by asking, “What happened to the input to get the output?”
2. Write the rule as an expression.

Take a look at the following function rule and determine if it is a rule for the data in the table below.

x+4\begin{align*}x+4\end{align*}

 Input Output 2 5 3 6 4 7 5 8

First, substitute the input values in for x\begin{align*}x\end{align*} to see if you get the corresponding output value.

x+42+46\begin{align*}& x + 4\\ & 2 + 4\\ & 6\end{align*}

This does not equal the corresponding output value of 5.

Look at the other input values. Each term in the input became the term in the output when 3 was added to it. The rule states that four was added. Therefore, this is not a viable rule.

Here is another function.

5x\begin{align*}5x\end{align*}

Determine if 5x\begin{align*}5x\end{align*} is a function rule for the data in the table below.

 Input Output 20 100 10 50 5 25 1 5

First, substitute the input values in for x\begin{align*}x\end{align*} to see if you get the corresponding output value.

5x5(20)100\begin{align*}& 5x\\ & 5(20)\\ & 100\end{align*}

Substitute another input value.

5x5(10)50\begin{align*}& 5x\\ & 5(10)\\ & 50\end{align*}

5x5(5)25\begin{align*}& 5x\\ & 5(5)\\ & 25\end{align*}

5x5(1)5\begin{align*}& 5x\\ & 5(1)\\ &5\end{align*}

So, yes it is. In this case, each term in the input was multiplied by five to get the term in the output. Therefore this rule does work for this table.

### Examples

#### Example 1

Earlier, you were given a problem about Geri and the change machine.

Geri knows that there are 4 quarters in a dollar, and that is why she put 6 dollars in the machine and received 24 quarters. How can Geri write this as a function rule?

This is a table to represent the change machine’s input and output.

 Input (dollars) Output (quarters) 6 24 5 20 4 16 3 12

First, look at the table and ask yourself, “What happened to x\begin{align*}x\end{align*} (input) to get y\begin{align*}y\end{align*} (output)?”

What happened to 5 to get 50? What happened to 6 to get 60 and so forth? If you look carefully, you will see that the input value (x)\begin{align*}(x)\end{align*} is multiplied by 10 to get the output value.

Next, use a variable for the input and write the rule.

You can write it as an expression, x(10)\begin{align*}x(10)\end{align*} or 10x\begin{align*}10x\end{align*}. This is the function rule, 10x\begin{align*}10x\end{align*}.

Then, see if the function rule 10x\begin{align*}10x\end{align*} works for each term in the table by plugging the input into the expression and seeing if it equals the listed output?

10x10(5)50\begin{align*}& 10x\\ & 10(5)\\ & 50\end{align*}

10x10(6)60\begin{align*}& 10x\\ & 10(6)\\ & 60\end{align*}

10x10(7)70\begin{align*}& 10x\\ & 10(7)\\ & 70\end{align*}

10x10(8)80\begin{align*}& 10x\\ & 10(8)\\ & 80\end{align*}

The answer is yes, this rule works for this table.

#### Example 2

Write a function rule to represent the data in this table.

 Input Output 3 5 5 9 7 13 8 15 10 19

First, look at the table and ask yourself, “What happened to x\begin{align*}x\end{align*} (input) to get y\begin{align*}y\end{align*} (output)?”

Here two operations were performed. The input value was multiplied by two and then one was subtracted.

Next, use a variable for the input and write the rule.

2(x)1\begin{align*}2(x) - 1\end{align*}

The answer or function rule is 2x1\begin{align*}2x - 1\end{align*}.

Then, see if the function rule 2x1\begin{align*}2x -1\end{align*} works for each term in the table by plugging the input into the expression and seeing if it equals the listed output.

Substitute the input values in for x\begin{align*}x\end{align*} in the function 2x1\begin{align*}2x -1\end{align*} to see if you get the results in the output column.

2x12(3)1615 (output)\begin{align*}& 2x - 1\\ & 2(3) - 1\\ & 6-1\\ & 5 \ (\text{output})\end{align*}

2x12(5)11019 (output)\begin{align*}& 2x - 1\\ & 2(5)- 1\\ & 10 - 1\\ & 9 \ (\text{output})\end{align*}

2x12(7)114113 (output)\begin{align*}& 2x - 1\\ & 2(7) - 1\\ & 14 - 1\\ & 13 \ (\text{output})\end{align*}

2x12(8)116115 (output)\begin{align*}& 2x - 1 \\ & 2(8) - 1\\ & 16 -1\\ & 15 \ (\text{output})\end{align*}

2x12(10)120119 (output)\begin{align*}& 2x - 1\\ & 2(10) - 1\\ & 20 - 1\\ & 19 \ (\text{output})\end{align*}

The answer is correct.

#### Example 3

Determine whether the following rule makes sense for the input-output table.

Rule: 4x\begin{align*}4x\end{align*}

 Input Output 2 10 3 15 5 25 6 30

First, let the input value be the variable x\begin{align*}x\end{align*}.

Next, substitute the input values in the expression for x\begin{align*}x\end{align*}.

4x4(2)8==10 (output)1010\begin{align*}\begin{array}{rcl} 4x &=& 10 \ (\text{output})\\ 4(2) &=& 10\\ 8 & \neq &10 \end{array}\end{align*}

The answer is no, this rule does not work for this table.

#### Example 4

Determine whether the following rule makes sense for the input-output table.

Rule: \begin{align*}2x - 1\end{align*}

 Input Output 2 3 3 5 4 7 6 11

First, let the input value be the variable \begin{align*}x\end{align*}.

Next, substitute the input values in the expression for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 2x - 1 &=& 3 \ (\text{output})\\ 2(2) - 1 &=& 3\\ 4 -1 &=& 3\\ 3 &=& 3 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} 2x -1 &=& 5\\ 2(3) - 1 &=& 5\\ 6 -1 &=& 5\\ 5 &=& 5 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} 2x - 1 &=& 7\\ 2 (4) - 1 &=& 7\\ 8 - 1 &=& 7\\ 7 &=& 7 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} 2x - 1 &=& 11\\ 2(6) - 1 &=& 11\\ 12 - 1 &=& 11\\ 11 &=& 11 \end{array}\end{align*}

The answer is yes, this rule does work for this table.

#### Example 5

Determine whether the following rule makes sense for the input-output table.

Rule: \begin{align*}3x\end{align*}

 Input Output 2 6 3 9 4 12 6 18

First, let the input value be the variable \begin{align*}x\end{align*}.

Next, substitute the input values in the expression for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 3x &=& 6 \ (\text{output})\\ 3(2) &=& 6\\ 6 &=& 6 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} 3x &=& 9\\ 3(3) &=& 9\\ 9 &=& 9 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} 3x &=& 12\\ 3(4) &=& 12\\ 12 &=& 12 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} 3x &=& 18\\ 3(6) &=& 18\\ 18 &=& 18 \end{array}\end{align*}

The answer is yes, this rule does work for this table.

### Review

Evaluate each given function rule to determine if the rule works for the data in the table.

1. \begin{align*}2x+2\end{align*}
 Input Output 2 6 3 8 4 10 5 12
1. \begin{align*}3x\end{align*}
 Input Output 1 4 2 6 3 10
1. \begin{align*}5x+1\end{align*}
 Input Output 1 6 2 11 3 16 4 21
1. \begin{align*}2x\end{align*}
 Input Output 1 3 2 5 3 7
1. \begin{align*}3x-1\end{align*}
 Input Output 1 2 2 5 3 8 4 11
1. \begin{align*}2x+1\end{align*}
 Input Output 1 3 2 4 3 6 5 10
1. \begin{align*}4x\end{align*}
 Input Output 0 0 1 4 2 8 3 12
1. \begin{align*}6x-3\end{align*}
 Input Output 1 3 2 9 3 15
1. \begin{align*}2x\end{align*}
 Input Output 0 0 1 2 2 4 3 6
1. \begin{align*}3x-3\end{align*}
 Input Output 1 0 2 3 4 9 5 12

Create a table for each rule.

1. \begin{align*}5x\end{align*}
2. \begin{align*}6x+1\end{align*}
3. \begin{align*}2x-3\end{align*}
4. \begin{align*}3x+3\end{align*}
5. \begin{align*}4x+1\end{align*}

To see the Review answers, open this PDF file and look for section 12.11.

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### Vocabulary Language: English

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.

Input-Output Table

An input-output table is a table that shows how a value changes according to a rule.

Pattern

A pattern is a series of pictures, numbers or other symbols that repeat in some way according to a rule.