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# Input-Output Tables for Function Rules

## Use function rules to complete patterns in tables.

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Input-Output Tables for Function Rules

Soren is working on a project about the eating behavior of the Mandarin duck from East Asia. He has gathered some data about the Mandarin duck’s weight and the approximate amount of plant matter it eats in a day. Soren put this information in a table and noticed a pattern in the duck’s weight and the approximate amount of plant matter eaten.

 Input (Mandarin duck weight in grams) Output (Plant matter eaten in grams) 400 250 500 300 600 350 700 400

Can you write a rule that describes the data in this table?

In this concept, you will learn to evaluate a given function rule using an input-output table.

### Using Input-Output Tables for Function Rules

An input-output table, like the one shown below, can be used to represent a function.

 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.

Now look at how to 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.

 Input number (\begin{align*}x\end{align*}) Output number (\begin{align*}y\end{align*}) 0 1 2.5 5 10

To find each 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+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+1.5=6.5\end{align*} 10 11.5 \begin{align*}\leftarrow 10+1.5=11.5\end{align*}

You can represent the information in this table as five ordered pairs from the function.

Now, write the answer in ordered pairs.

The answer is \begin{align*}(0, 1.5) (1, 2.5) (2.5, 4) (5, 6.5) (10, 11.5)\end{align*}.

Let’s look at another example to show how to create a function table given a rule.

Let’s say that the domain of a function is all real numbers. The rule for this function is: multiply each \begin{align*}x\end{align*}-value (the input) by 4 and then subtract 2. Make a table that shows three inputs and three corresponding outputs for a function that follows this rule. Then, represent this information as ordered pairs.

First, choose three \begin{align*}x\end{align*}-values for the table. You may choose any numbers in the domain of the function, but let’s select some numbers that are easy to work with such as 1, 2, and 3.

Next, you take an input value (\begin{align*}x\end{align*}-value), and apply the function rule to find the corresponding output value (\begin{align*}y\end{align*}-value).

The rule for this function is: multiply each \begin{align*}x\end{align*}-value (the input) by 4 and then subtract 2.

 Input number (\begin{align*}x\end{align*}) Output number (\begin{align*}y\end{align*}) 1 2 \begin{align*}\leftarrow 4(1)-2=2\end{align*} 2 6 \begin{align*}\leftarrow 4(2)-2=6\end{align*} 3 10 \begin{align*}\leftarrow 4(3)-2=10\end{align*}

You can represent the information in this table (the inputs and outputs of the function) as three ordered pairs.

The answer is \begin{align*}\{(1,2),(2,6),(3,10)\}\end{align*}.

You can also write the rule for a function rule as an equation.

Here is an example.

The equation \begin{align*}y = \frac{x}{3} + 1\end{align*} is a function. The input is the \begin{align*}x\end{align*}-value and the corresponding \begin{align*}y\end{align*}-value is the output. Use this rule to find the missing \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values in the table below.

 Input number (\begin{align*}x\end{align*}) Output number (\begin{align*}y\end{align*}) 0 1 3 2 9 \begin{align*}\Box{}\end{align*} \begin{align*}\Box{}\end{align*} 8

First, when \begin{align*}x\end{align*} is 9, use the function to solve for \begin{align*}y\end{align*}. You do this by substituting 9 in for \begin{align*}x\end{align*} in \begin{align*}y = \frac{x}{3} + 1\end{align*}.

\begin{align*}\begin{array}{rcl} y &=& \frac{x}{3}+1\\ y &=& \frac{9}{3}+1\\ y &=& 3 + 1\\ y &=& 4 \end{array}\end{align*}

One answer is when \begin{align*}x = 9, y = 4\end{align*}. This means that \begin{align*}(9, 4)\end{align*} is an ordered pair for this function.

Next, solve for the other missing value.

You know that \begin{align*}y=8\end{align*}. So, plug that into the function and solve for \begin{align*}x\end{align*}

\begin{align*}\begin{array}{rcl} y &=& \frac{x}{3} + 1\\ 8 &=& \frac{x}{3} + 1 \end{array}\end{align*}

Then, use the subtraction property of equality and subtract 1 from both sides of the equation.

\begin{align*}\begin{array}{rcl} 8-1 &=& \frac{x}{3} + 1-1\\ 7 &=& \frac{x}{3} \end{array}\end{align*}

Next, use the multiplication property of equality and multiply both sides of the equation by 3.

\begin{align*}\begin{array}{rcl} 7 \times 3 &=& \frac{x}{3} \times 3\\ 21 &=& \frac{3}{3}x\\ 21 &=& 1x\\ 21 &=& x \end{array}\end{align*}

The answer is when the output value (\begin{align*}y\end{align*}) is 8 the input value (\begin{align*}x\end{align*}) is 21. This means that \begin{align*}(21, 8)\end{align*} is an ordered pair for this function.

### Examples

#### Example 1

Earlier, you were given a problem about Soren and the information he’s gathering about Mandarin ducks.

Soren notices a pattern in the information he’s collected so far. Can you write a rule that takes the input and gives the output number?

 Input (Mandarin duck weight in grams) Output (Plant matter eaten in grams) 400 250 500 300 600 350 700 400

Let \begin{align*}x\end{align*} be the input, and \begin{align*}y\end{align*} be the output.

You can notice that the outputs are about half the inputs, and as you look closer you may notice that the outputs are half the input plus 50. That is, the output, \begin{align*}y\end{align*}, is \begin{align*}\frac{x}{2}+50 \end{align*}.

You can write this as a function taking an input value (\begin{align*}x\end{align*}) to an output value (\begin{align*}y\end{align*}).

\begin{align*}y = \frac{x}{2}+50\end{align*}

The answer for a rule or a function which takes duck weight as an input and gives plant matter eaten as an output is \begin{align*}y = \frac{x}{2}+50\end{align*}.

#### Example 2

Data about the number of chipmunks and the corresponding number of nuts eaten is presented in the following table.

 Input (number of chipmunks) Ouptut (nuts eaten) 12 6 10 5 8 4 6 3 4 2

Can you write a rule for a function that describes this information?

First, examine the inputs and outputs to see if there is a clear pattern. You can see that if you divide the input by 2 you get the output.

Next, let \begin{align*}x\end{align*} be the input, and \begin{align*}y\end{align*} be the output. You can then write a rule that describes this function.

The answer is \begin{align*}y = \frac{x}{2}\end{align*}.

Solve each equation.

#### Example 3

Complete the table given the following rule: Add 2 to the input to get the output.

 Input (\begin{align*}x\end{align*}) Output (\begin{align*}y\end{align*}) 3 5 6

First, add 2 to each input to find the output.

The answers are 5, 7 and 8.

#### Example 4

Write an equation that describes the function in Example 1.

Use the rule: add 2 to the input to get the output. Substitute \begin{align*}x\end{align*} for the input, and \begin{align*}y\end{align*} for the output.

The answer is \begin{align*}y = x + 2\end{align*}.

#### Example 5

Create a table for the function \begin{align*}y = \frac{x}{2} + 3\end{align*}. Use three values for the inputs.

First, it is a good idea to pick numbers for the inputs that will be easy to calculate. Let’s take the numbers 2, 4, and 6 since they are all even numbers divisible by 2.

Next, plug each input value into \begin{align*}x\end{align*}, in the function \begin{align*}y = \frac{x}{2} + 3\end{align*} and solve for \begin{align*}y\end{align*}.

 Input (\begin{align*}x\end{align*}) Output (\begin{align*}y\end{align*}) 2 4 4 5 6 6

### Review

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. Use this rule to find the corresponding output numbers.
 Input number (\begin{align*}x\end{align*}) Output number (\begin{align*}y\end{align*}) 0 1 2 3 4
1. 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.
 Input number (\begin{align*}x\end{align*}) Output number (\begin{align*}y\end{align*}) 0 6 7 16
1. The equation \begin{align*}y = \frac{x}{2}-1\end{align*} is a function. Use this function to find the missing values in the table below.
 Input number (\begin{align*}x\end{align*}) Output number (\begin{align*}y\end{align*}) 2 0 4 1 8 6

Determine if the rule provided would give the information in the input-output table. If it works write “yes if not, write “no”.

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

Create a table for each rule. Use five input values.

1. \begin{align*}7x\end{align*}
2. \begin{align*}3x+1\end{align*}
3. \begin{align*}5x-3\end{align*}
4. \begin{align*}4x+3\end{align*}
5. \begin{align*}4x-5\end{align*}

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Color Highlighted Text Notes

### Vocabulary Language: English

domain

The domain of a function is the set of $x$-values for which the function is defined.

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 $y$.

Function Rule

A function rule describes how to convert an input value ($x$) into an output value ($y$) for a given function. An example of a function rule is $f(x) = x^2 + 3$.

Function Table

A function table is another name for an input-output table, a table that shows how a value changes according to a rule.

Input-Output Table

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

Linear Function

A linear function is a relation between two variables that produces a straight line when graphed.

Range

The range of a function is the set of $y$ values for which the function is defined.