What if you were given a table of *x* and *y* values? How could you write a rule to describe the relationship between the two variables? After completing this Concept, you'll be able to write a function rule for tables like this.

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CK-12 Foundation: 0111S Write a Function Rule

### Guidance

In many situations, we collect data by conducting a survey or an experiment, and then organize the data in a table of values. Most often, we want to find the function rule or formula that fits the set of values in the table, so we can use the rule to predict what could happen for values that are not in the table.

#### Example A

*Write a function rule for the following table:*

\begin{align*}& \text{Number of CDs} \qquad 2 \qquad 4 \qquad 6 \qquad 8 \qquad 10\\ & \text{Cost in} \ \$ \qquad \qquad \ \ 24 \quad \ \ 48 \quad \ \ 72 \quad \ 96 \quad \ \ 120\end{align*}

**Solution**

You pay $24 for 2 CDs, $48 for 4 CDs, $120 for 10 CDs. That means that each CD costs $12.

We can write a function rule:

Cost \begin{align*}= \$12 \ \times\end{align*}

#### Example B

*Write a function rule for the following table:*

\begin{align*}& x \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad 2 \quad 3\\ & y \qquad 3 \ \qquad 2 \qquad \ 1 \quad 0 \quad 1 \quad 2 \quad 3\end{align*}

**Solution**

You can see that a negative number turns into the same number, only positive, while a non-negative number stays the same. This means that the function being used here is the absolute value function: \begin{align*}f(x) = \mid x \mid\end{align*}

Coming up with a function based on a set of values really is as tricky as it looks. There’s no rule that will tell you the function every time, so you just have to think of all the types of functions you know and guess which one might be a good fit, and then check if your guess is right. In this book, though, we’ll stick to writing functions for linear relationships, which are the simplest type of function.

**Represent a Real-World Situation with a Function**

Let’s look at a few real-world situations that can be represented by a function.

#### Example C

*Maya has an internet service that currently has a monthly access fee of $11.95 and a connection fee of $0.50 per hour. Represent her monthly cost as a function of connection time.*

**Solution**

**Define**

Let \begin{align*}x =\end{align*}

Let \begin{align*}y =\end{align*}

**Translate**

The cost has two parts: the one-time fee of $11.95 and the per-hour charge of $0.50. So the total cost is the flat fee \begin{align*}+\end{align*}

**Answer**

The function is \begin{align*}y = f(x) = 11.95 + 0.50x\end{align*}

#### Example D

*Alfredo wants a deck build around his pool. The dimensions of the pool are \begin{align*}12 \ feet \times 24 \ feet\end{align*} 12 feet×24 feet and the decking costs $3 per square foot. Write the cost of the deck as a function of the width of the deck.*

**Solution**

**Define**

Let \begin{align*}x =\end{align*}

Let \begin{align*}y =\end{align*}

**Make a sketch and label it**

**Translate**

You can look at the decking as being formed by several rectangles and squares. We can find the areas of all the separate pieces and add them together:

\begin{align*}\text{Area} &= 12x + 12x + 24x + 24x + x^2 + x^2 + x^2 + x^2 \\ &= 72x + 4x^2\end{align*}

To find the total cost, we then multiply the area by the cost per square foot ($3).

**Answer**

\begin{align*}f(x) = 3(72x + 4x^2) = 216x + 12x^2\end{align*}

Watch this video for help with the Examples above.

CK-12 Foundation: Write a Function Rule

### Vocabulary

- Most often, we want to find the
**function rule**or formula that fits the set of values in the table, so we can use the rule to**predict**what could happen for values that are not in the table.

### Guided Practice

*A cell phone company sells two million phones in their first year of business. The number of phones they sell doubles each year. Write a function that gives the number of phones that are sold per year as a function of how old the company is.*

**Solution**

**Define**

Let \begin{align*}x =\end{align*} age of company in years

Let \begin{align*}y =\end{align*} number of phones that are sold per year

**Make a table**

\begin{align*}& \text{Age (years)} \qquad \qquad 1 \quad 2 \quad 3 \quad 4 \quad \ 5 \quad \ \ 6 \quad \ \ 7\\ & \text{Millions of phones} \quad 2 \quad 4 \quad 8 \quad 16 \quad 32 \quad 64 \quad 128\end{align*}

**Write a function rule**

The number of phones sold per year doubles every year, so the first year the company sells 2 million phones, the next year it sells \begin{align*}2 \times 2\end{align*} million, the next year it sells \begin{align*}2 \times 2 \times 2 \end{align*} million, and so on. You might remember that when we multiply a number by itself several times we can use exponential notation: \begin{align*}2 = 2^1, \ 2 \times 2 = 2^2, \ 2 \times 2 \times 2 = 2^3,\end{align*} and so on. In this problem, the exponent just happens to match the company’s age in years, which makes our function easy to describe.

**Answer**

\begin{align*}y = f(x) = 2^x\end{align*}

### Practice

- Write a function rule for the following table: \begin{align*}& x \quad 3 \quad 4 \quad \ \ 5 \quad \ 6\\
& y \quad 9 \quad 16 \quad 15 \quad 36\end{align*}
- Write a function rule for the following table: \begin{align*}& \text{Hours} \quad 0 \quad \ 1 \quad \ 2 \quad \ \ 3\\
& \text{Cost} \quad \ 15 \quad 20 \quad 25 \quad 30\end{align*}
- Write a function rule for the following table: \begin{align*}& x \quad 0 \quad \ \ 1 \quad \ 2 \quad 3\\
& y \quad 24 \quad 12 \quad 6 \quad 3\end{align*}
- Write a function that represents the number of cuts you need to cut a ribbon into \begin{align*}x\end{align*} pieces.
- Write a function that represents the number of cuts you need to divide a pizza into \begin{align*}x\end{align*} slices.

For 6-8, suppose Solomon charges a $40 flat rate plus $25 per hour to repair a leaky pipe.

- Write a function that represents the total fee charged as a function of hours worked.
- How much does Solomon earn for a 3-hour job?
- How much does he earn for three separate 1-hour jobs?

For 9-12, suppose Rochelle has invested $2500 in a jewelry making kit. She makes bracelets that she can sell for $12.50 each.

- Write a function that shows how much money Rochelle makes from selling \begin{align*}b\end{align*} bracelets.
- Write a function that shows how much money Rochelle has after selling \begin{align*}b\end{align*} bracelets, minus her investment in the kit.
- How many bracelets does Rochelle need to make before she breaks even?
- If she buys a $50 display case for her bracelets, how many bracelets does she now need to sell to break even?