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Function Rules for Input-Output Tables

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Writing Function Rules
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Have you ever eaten a churro? Take a look at this dilemma.

You are going to order a bunch of churros from a vender for your family. Each one costs $1.50. How much will they cost if you buy 6 or 8 or 10? What’s the function rule?

This Concept will show you how to write a function rule for a situation like this one.

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 x coordinate in the relation. The range is made up of the second column or the y 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.

There are function rules that go with function tables. Do you know what a function rule is?

A function rule 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.

We can also use input/output tables to help us to write function rules. When we look at the input and decipher what happened to it to create the output, then we can write a function rule based on our discoveries. This is like being a detective! You will have to use what you have learned and look for clues.

Take a look.

x f(x)
0 5
3 8
6 11
9 14

The first thing to notice is that the words input and output have been replaced by the x and the f(x) . This means that we are using function notation to say that x is the input and that the output is a function of x . That is what the f(x) means.

That is exactly what you need to do.

In looking at this pattern, you can see that the x value is increased in each step of the table. Each x value is increased by 5. This means that we can write the following rule for our function.

f(x) = x + 5

This is our answer.

Write down an example of function notation and that you need to look for a pattern when figuring out function rules. Put this information in your notebook.

Write a function rule for each example.

Example A

x f(x)
9 8
11 10
15 14
17 16

Solution: x-1

Example B

x f(x)
3 7
9 19
10 21
12 25

Solution: 2x+1

Example C

x f(x)
9 -27
11 -33
15 -45
16 -48

Solution: -3x

Now let's go back to the dilemma at the beginning of the Concept.

First, we can take the given information to write the rule.

p(c)=1.50c where p is total price and c is the number of churros.

Next, we can we can substitute different values into the function rule to figure out the cost for 6, 8 or 10 churros.

 & p(c) =1.50c && p(c)=1.50c && p(c) =1.50c\\& p(6) =1.50 \cdot 6 && p(8)=1.50 \cdot 8 && p(10)=1.50 \cdot 10\\& p(6) =9 && p(8) =12 && p(10)=15\\& \$ 9 \text{ for } 6 \text{ churrors} && \$ 12 \text{ for } 8 \text{ churros} && \$ 15 \text{ for } 10 \text{ churros}

Based on the number of churros, we can figure out the differences in cost. This is our answer.

Vocabulary

Relation
a set of ordered pairs.
Domain
the x value in a table or function.
Range
the y 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 x value or the domain of a function.
Output
the y value or the range of a function.

Guided Practice

Here is one for you to try on your own.

Use function notation to write a function rule for the given table.

x f(x)
12 6
9 4.5
7 3.5
4 2

Solution

Let's start by looking for a pattern. Do you notice one?

Each value of the input has been divided in half. We can write this function rule in two ways.

f(x)&= \frac{1}{2}x \\or \ f(x)&= \frac{x}{2}

Both of these will work as a correct answer.

Video Review

Writing Function Rules

Practice

Directions: Write function rules.

  1. Write a function rule for the following data.
x f(x)
9 27
11 33
15 45
16 48
  1. Write a function rule for the following data:
x f(x)
-2 -6
0 -4
2 -2
4 0
  1. Write a function rule for the following data:
x f(x)
0 0
1 2
4 8
5 10
  1. Write a function rule for the following table.
x f(x)
1 0
2 2
4 6
8 14
  1. Write a function rule for the following table.
x f(x)
2 1
4 2
8 4
10 5
18 9
  1. Write a function rule for each table.
x f(x)
6 2
9 3
15 5
21 7
30 10
  1. Write a function rule for each table.
x f(x)
2 3
9 10
15 16
21 22
30 31
  1. Write a function rule for each table.
x f(x)
3 -6
9 -18
15 -30
20 -40
24 -48

Directions: Solve the following problem.

  1. Sandwich cost $3.45 each. Write a function rule for the cost, c , for a number of sandwiches, s .
  2. Now, find the cost of 3 sandwiches.
  3. Find the cost of 6 sandwiches.
  4. Find the cost of 9 sandwiches.
  5. Find the cost of 2 sandwiches.
  6. Find the cost of 8 sandwiches.
  7. Find the cost of a dozen sandwiches.

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