Lesson 4 Composing and Decomposing Develop Understanding

Learning Focus

Examine function composition in the context of a real-world scenario.

Are there ways to combine functions other than by adding, subtracting, multiplying, or dividing them?

How do I model contexts in which one function depends upon the output of another function?

Open Up the Math: Launch, Explore, Discuss

As the day at the amusement park gets warmer, you and your friends decide to cool off by taking a ride on the Turbulent Waters Dive. As you are waiting in line your tour guide explains the mathematics behind designing the waiting area for a ride.

“As you can see,” says the engineer, “the waiting area can be enlarged or reduced by moving a few chains around. The area we need for waiting guests depends on the time of day. We collect data for each ride so we can use functions to model the typical wait time and how much waiting area we need to provide for our guests.”

And of course, your guide has the functions that represent this particular ride.

• Average number of people in the TWD line as a function of time: $p (t) = 3,000 (\cos \frac{1}{5} (t - 3))$

$t$ is the number of hours before or after noon, so $t = 2$ represents 2:00 p.m. and $t = - 2$ represents 10:00 a.m.
$p$ represents the number of people in line

• Waiting area required as a function of the number of people in line: $a (p) = 4 p + 100$

$a$ , the waiting area, is measured in square feet

• Wait time for a guest as a function of the number of people in line: $w (p) = 60 \cdot (\frac{p - 1, 500}{1, 500})$

$w$ , the wait time, is measured in minutes

Interpreting the Functions

1.

At what time of day is the number of people in line the largest?

2.

What is the maximum number of people in line, based on the function for the average number of people in line?

3.

When do you think the amusement park opens and closes, based on this function?

4.

In terms of the story context, what do you think the $4$ and the $100$ represent in function rule for waiting area, $a (p) = 4 p + 100$ ?

5.

In terms of the story context, what might be the meaning of the $1,500$ in the function rule for wait time, $w (p) = 60 \cdot (\frac{p - 1, 500}{1, 500})$ ?

6.

How much waiting area is required for the guests in line for the Turbulent Waters Dive at each of the times listed in the following table?

a.

Time of Day	Waiting Area Required (sq. ft.)
10:00 a.m.
12:00 noon
2:00 p.m.
4:00 p.m.
8:00 p.m.

b.

For each instant in time, you had to complete a series of calculations. Describe how you found the waiting area at different times.

c.

Can you create a single rule that will determine the waiting area as a function of the time of day?

7.

What is the wait time for a guest that arrives at the end of the line for the Turbulent Waters Dive at each of the times listed in the following table?

a.

Time of Day	Wait Time (minutes)
10:00 a.m.
12:00 noon
2:00 p.m.
4:00 p.m.
8:00 p.m.

b.

For each instant in time, you had to complete a series of calculations. Describe how you found the wait time at different times of the day.

c.

Can you create a single rule that will determine the wait time as a function of the time of day?

Pause and Reflect

To maintain crowd control when the lines get long, cast members dressed as pirates (the Turbulent Waters Dive has a pirate theme) mingle with the waiting guests. Their antics distract the guests who listen to their pirate jokes. The number of cast members needed depends on the number of people waiting in the line.

Number of cast members needed as a function of the number of people in line: $c (p) = \frac{p}{150}$
- $p$ represents the number of people in line
- $c$ represents the number of cast members needed

8.

How many cast members are needed to entertain and distract the waiting guests at each of the following times of the day?

Time of Day	Cast Members Needed
10:00 a.m.
12:00 noon
2:00 p.m.
4:00 p.m.
8:00 p.m.
$t$ hours before or after noon ( $t < 0$ before noon, $t > 0$ after noon)

On warm, sunny days, misters are used to cool down the waiting guests. The number of misters that need to be turned on depends on the size of the waiting area that has been opened up to contain the number of people in line.

Number of misters needed as a function of the waiting area: $m (a) = \frac{a}{1,000}$
- $a$ , the waiting area, is measured in square feet
- $m$ represents the number of misters to be turned on

9.

How many misters need to be turned on to cool the waiting guests at each of the following times of day?

Time of Day	Misters Needed
10:00 a.m.
12:00 noon
2:00 p.m.
4:00 p.m.
8:00 p.m.
$t$ hours before or after noon ( $t < 0$ before noon, $t > 0$ after noon)

10.

Explain how the following diagram might help you think about the work you have been doing on the previous problems. How does the notation used in the diagram support the way you have been combining functions in this task? This way of combining functions is called function composition.

Interpreting the Functions

11.

In terms of the story context, what might be the meaning of the $150$ in the function rule for cast members needed, $c (p) = \frac{p}{150}$ ?

12.

In terms of the story context, what might be the meaning of the $1,000$ in the function rule for the number of misters needed, $m (a) = \frac{a}{1,000}$ ?

Ready for More?

Is function composition commutative? That is, does the order matter for which function we evaluate first and which we evaluate second? Illustrate your answer to this question using examples built from the given functions in this task.

Takeaways

While we can combine functions by adding, subtracting, multiplying, or dividing them, we can also combine functions using function composition. To compose two functions, .

The operation of function composition can be represented by or by .

Function composition can be used to create models for . For example,

We needed to use function composition in the Wait Time equation $w (t) = 60 (\frac{3, 000 \cos (\frac{1}{5} (t - 3)) - 1, 500}{1, 500})$ , since the wait time is a function of .
We needed to use function composition in Ferris wheel equations like $h (t) = 25 \sin (18 t) + 30$ , since the distance above or below the center of the wheel was a function of , and the angle of rotation was a function of .

Vocabulary

composition of functions
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned a new operation for combining functions called function composition. We use function composition when the input for one function is the output of another function.

Retrieval

1.

The expression $5 x^{3}$ contains $2$ operations. One of the operations is “inside” the second operation. Identify the “inside” operation as $u$ by writing . Then substitute $u$ into the expression so that the “outside” operation is being performed on $u$ . Write your answer by filling in the blanks on the If … then … statement.

If , then .

2.

Would the answer in problem 1 have been different if you were given ${(5 x)}^{3}$ ? Explain.

3.

Change each logarithmic expression to an equivalent expression involving an exponent.

a.

$\log_{2} (\frac{1}{4}) = - 2$

b.

$\log 873 = N$

c.

$\ln \sqrt{e} = \frac{1}{2}$