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Functions that Describe Situations

Compose functions to describe story problems

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Functions that Describe Situations

What if your bank charges a monthly fee of $15 for your checking account and also charges $0.10 for each check written? How would you represent this scenario with a function? Also, what if you could only afford to spend $20 a month on fees? Could you use your function to find out how many checks you could write per month?

Writing Functions

Functions from Tables

In many situations, data is collected by conducting a survey or an experiment. To visualize the data, it is arranged into a table. Often, you can examine the data in the table and determine a pattern or rule that describes the relationship between the sets of values.

Let's write a function that represents each of the tables below:

  1. \begin{align*}& \text{Number of CDs} && 2 && 4 && 6 && 8 && 10\\ & \text{Cost}\ (\$) && 24 && 48 && 72 && 96 && 120\end{align*}Number of CDsCost ($)22444867289610120

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

We can write the function rule.

\begin{align*}\text{Cost} = \$12 \times \text{number of CDs}\end{align*}Cost=$12×number of CDs or \begin{align*}f(x) = 12x\end{align*}f(x)=12x

 

  1. \begin{align*}&& x && -3 && -2 && -1 && 0 && 1 && 2 && 3\\ && y && \quad 3 && \quad 2 && \quad 1 && 0 && 1 && 2 && 3\end{align*}xy33221100112233

The values of the dependent variable are always the corresponding positive outcomes of the input values. This relationship has a special name, the absolute value. The function rule looks like this: \begin{align*}f(x) = |x|\end{align*}f(x)=|x|.

Functions from Descriptions

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

Let's write a function that represents the following situation:

  1. 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.

Let \begin{align*}x=\end{align*}x= the number of hours Maya spends on the internet in one month, and let \begin{align*}y=\end{align*}y= Maya’s monthly cost. The monthly fee is $11.95 with an hourly charge of $0.50.

The total cost \begin{align*}=\end{align*}= flat fee \begin{align*}+\end{align*}+ hourly fee \begin{align*}\times\end{align*}× number of hours. The function is \begin{align*}y = f(x) = 11.95 + 0.50x\end{align*}y=f(x)=11.95+0.50x.

 

 

 

 

Examples

Example 1

Earlier, you were asked to represent the following scenario as a function: Your bank charges you a monthly fee of $15 for your checking account and also charges you $0.10 for each check written. 

Let \begin{align*}x=\end{align*}x= the number of checks you write in a month and let \begin{align*}f(x)\end{align*}f(x) be your monthly costs at the bank. The total cost will be the monthly fee \begin{align*}+\end{align*}+ the check fee \begin{align*}\times\end{align*}× the number of checks you write. Thus, the function to represent this situation is:

\begin{align*}f(x)=15 + 0.1x\end{align*}f(x)=15+0.1x 

If you could only spend $20 a month on fees, then you would need to find the value of \begin{align*}x\end{align*}x that gives 20 as  \begin{align*}f(x)\end{align*}f(x). In this case, that number would be 50 since:

\begin{align*}20=15+0.1x\\ 20=15+0.1(50)\\ 20=15+5\\ 20=20\end{align*}20=15+0.1x20=15+0.1(50)20=15+520=20

For Examples 2-3, use the following information: When diving in the ocean, you must consider how much pressure you will experience from diving a certain depth. From the atmosphere, we experience 14.7 pounds per square inch (psi) and for every foot we dive down into the ocean, we experience another 0.44 psi in pressure.

Example 2

Write a function expressing how pressure changes depending on depth underwater.

We are always experiencing 14.7 psi from the atmosphere, and that pressure increases by 0.44 psi for every foot we descend into the ocean. Let \begin{align*}d\end{align*}d be our depth in feet underwater. Our dependent variable is the pressure \begin{align*}P\end{align*}P, which is a function of \begin{align*}d\end{align*}d:

\begin{align*}P=P(d)=14.7+0.44d\end{align*}

Example 3

How far can you dive without experiencing more than 58.7 psi of pressure on your body?

We want to know what our depth would be for a pressure of 58.7 psi.

\begin{align*}58.7=14.7+0.44d\end{align*}

Simplifying our equation by subtracting 14.7 from each side:

\begin{align*}44=0.44d\end{align*}

What should \begin{align*}d\end{align*} be in order to satisfy this equation? It looks like \begin{align*}d\end{align*} should be 100. Let's check:

\begin{align*}44=0.44(100)=44\end{align*}

So we do not want to dive more than 100 feet, because then we would experience more than 58.7 psi of pressure.

Review

  1. Use the following situation: Sheri is saving for her first car. She currently has $515.85 and is saving $62 each week.
    1. Write a function rule for the situation.
    2. Can the domain be “all real numbers"? Explain your thinking.
    3. How many weeks would it take Sheri to save $1,795.00?
  2. Write a function rule for the table.

\begin{align*}&& x && 3 && 4 && 5 && 6\\ && y && 9 && 16 && 25 && 36\end{align*}

  1. Write a function rule for the table.

\begin{align*}&\text{hours} && 0 && 1 && 2 && 3\\ &\text{cost} && 15 && 20 && 25 && 30\end{align*}

  1. Write a function rule for the table.

\begin{align*}&& x && 0 && 1 && 2 && 3\\ && y && 24 && 12 && 6 && 3\end{align*}

  1. Write a function that represents the number of cuts you need to cut a ribbon in \begin{align*}x\end{align*} number of pieces.
  2. Solomon charges a $40 flat rate and $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 three-hour job?
  3. Rochelle has invested $2500 in a jewelry-making kit. She makes bracelets that she can sell for $12.50 each. How many bracelets does Rochelle need to make before she breaks even?

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.12.

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Vocabulary

dependent variable

A dependent variable is one whose values depend upon what is substituted for the other variable.

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.

independent variable

The independent variable is the variable which is not dependent on another variable. The dependent variable is dependent on the independent variable.

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.

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