How to write a function rule

In this lesson, you will learn to write a function rule using information given in a table.

How to write a function rule
A function rule such as cost = p + 0.08p is an equation that describes a functional relationship.

If p is the price you pay for an item and 0.08 is the sales tax, the function rule above is the cost of the item.

If you are given a table, usually you have to carefully examine the table to see what the function rule is.

Sometimes, it is easy to get the rule. Other times, it is not so easy. It may require some keen observation or using the concept of slope.

We will now show how to do this with some examples:

Example #1:

Suppose you are washing your clothes at a laundromat.The table below shows the relationship between number of loads and cost.

Write a function rule showing the relationship between number of loads and cost.

Number of loads 1 2 3 4 5
Cost 3 6 9 12 15


First, show the relationship:

Cost is 3 × number of loads. For instance, 2 × 3 = 6

Second, define:

Let n be the number of loads

Let c be the cost.

Write the equation:

c = 3 × n

Example #2:

Write a function rule for the following situation:

Input 1 2 3 4 5
Output 5 6 7 8 9


First, show the relationship:

Output is 4 + input.

Second, define:

Let I = input

Let O = output.

Write the equation:

O = 4 + I

A more complicated example showing how to write a function rule.

Example #3:

Number of items sold
Revenue
1
6
2
10
3
14
4
18
5
22


First, show the relationship:

It is not that easy to see what the relationship is in this case.

1 × 6 = 6. However, 2 × 6 is not equal to 10.

1 + 5 = 6. However, 2 + 5 is not equal to 10.

The function rule could be a combination of addition and multiplication.

You could pick a revenue say 10 for instance and ask yourself "What math should I do with the 2 to get this revenue?"

Trial and error can help.

My strategy of trial and error is shown below:

2 × 1 + 8 = 10. However, 3 × 1 + 8 is not equal to 14. This rule does not work

2 × 2 + 6 = 10. However, 3 × 2 + 6 is not equal to 14. This rule does not work

2 × 3 + 4 = 10. However, 3 × 3 + 4 is not equal to 14. This rule does not work

2 × 4 + 2 = 10. Moreover, 3 × 4 + 2 is also equal to 14. This rule seems work! In fact, it will work

The relationship is then revenue is number of items × 4 plus two.

Second, define:

Let x be the number of items

Let R be the revenue

Write the equation:

R = x × 4 + 2

How to write a function rule using the concept of slope.

Let us do this for example #3. The goal is use the equation y = mx + b.

Using (1,6) and (2,10), m = (10 - 6) / (2 - 1) = 4 / 1 = 4

y = 4x + b.

Use (1,6) to find b.

6 = 4(1) + b

6 = 4 + b.

Since 4 + 2 = 6, b = 2

y = 4x + 2

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