Alg2 Lesson 2.1: Intro to Function Composition

Lesson 2.1: Intro to

Function Composition

Obj: I can compose 2 functions given a variable.

EQ: How can I evaluate a function composition given
a numerical value?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

Poll

What will you do today?

• Facilitator

• Includer

• Resourcer

• Scribe

• I'll work by myself since I am way ahead/behind the rest of the class.

Part 1: Exploring Function

Composition in Context

Handout 2.1.A: A Ripple in the Pond

A stone tossed into a pond creates an expanding circular ripple. The radius of the circular ripple increases at a rate of 2 cm/sec. How can we express the area of the ripple as a function of time?

With a partner, complee the following tables with values for r(t), the radius after t seconds
(sec), and A(r), the area of the ripple with a radius of r centimeters (cm). Keep your answers
in terms of π.

Use your two tables to complete the following table:

Drag and Drop

What is the relationship between the radius length and the number of seconds since the stone hit the pond? The radius length is always ​

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times the number of seconds since the stone hit the pond.

What can you do to determine the area of the ripple if you are given the length of the radius? ​

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What is the relationship between the radius length and area?
The area is always the ​

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of the length of the radius multiplied by ​

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.

What could you do to determine the radius length when you are given the area of
the circular ripple? Divide the area by π and take the ​

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of the quotient.

Drag these tiles and drop them in the correct blank above

2

1/2

Yes

No

π

square

square root

Filled in Table

Change your responses as needed.

Drag and Drop

1. An equation for the length of the radius as a function of the time since the stone was tossed in the pond is r(t) = ​

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.

2. An equation for the area of the ripple as a function of the length of the radius is A(r) = ​

Placeholder

.

3. An equation for the area of the ripple as a function of the time since the stone hit the pond is A(r(t)) = (2t)²π = ​

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.

Drag these tiles and drop them in the correct blank above

2t

r²π

4t²π

4tπ

r³π

t²/2π

Multiple Choice

To determine the area of the ripple 3.25 seconds after the stone hits the water, what would you do first?

• Find the radius

• Find the height

• Convert seconds to minutes

Multiple Choice

What would be the radius of a ripple 3.25 seconds after the stone hits the water?

• 3.25 cm

• 7.5 cm

• 1.625 cm

• 10.5625 cm

Multiple Choice

What would be the area of a ripple 3.25 seconds after the stone hits the water?

• 56.25π cm²

• 15π cm²

• 24.5π cm²

• 10.5625π cm²

Multiple Select

What operations did you need to perform to determine the area of the ripples?

• Multiply time by 2

• Square the radius

• Multiply by π

• Multiply the radius by 2

• Square the time

Drag and Drop

What does each portion of your equation A(r(t)) = (2t)²π=4t²π represent in this scenario?
The 2t represents the ​

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of the radius of the ripple t ​

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after the stone hit the water. The 4t²π represents the ​

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of the ripple after 2t ​

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.

Drag these tiles and drop them in the correct blank above

length

seconds

area

centimeters

perimeter

The Pond Ripple is an Example of Composition

Composition is the term we use to describe
chaining together two or more functions to form one function. When we can express two functions
together with composition, we call it a composite
function.

In Your Journal…

1. Give an example of two real-world functions that

make sense to compose.

2. Does the order in which we compose functions

matter? Give an example to illustrate your answer.

Handout 2.1.B: Carpeting a Room

XYZ Flooring has been hired to install a carpet in a rectangular room. The
shorter dimension of the room is denoted by s and is measured in yards. The
longer dimension of the room is 1.5 times as long as the shorter dimension. The cost of installing the carpet is determined, in part, by the area to be carpeted; in addition to a flat service fee of $200, XYZ Flooring charges $3 per square yard of carpet. Find an algebraic representation of the cost in dollars, C, of installing carpet in the room.

Drag and Drop

What is the area of the room as a function of s?

A(s)=​

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What is the cost in dollars of installing carpeting as a function of the area of the room?

C(A) = ​

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What is the cost in dollars, C, of installing carpet as a function of s?

C(A(s))=​

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Drag these tiles and drop them in the correct blank above

(1.5(s))(s)=1.5s²

3A + 200

3(1.5s²)+200=4.5s²+200

(1.5(A))(A)=1.5A²

3s + 200

3(1.5A²)+200=4.5A²+200

Part 2: Composing

Functions Using a Table

Use the tables to determine the following values:

f(g(0))

f(g(1))

f(f(2))

g(g(0))

g(f(6))

g(f(1.25))

f(x)

g(x)

f(g(x))

g(f(x))

f(f(x))

g(g(x))

f(f(⅔))

g(g(-7))

To determine which table to use first to find the value of each composite function,
the innermost function tells us which table to
use first and the output of that function
becomes the input for the outermost function.

Multiple Choice

Evaluate f(g(0)).

• 4

• -5

• 3

• -1

• 0

Fill in the Blanks

Multiple Choice

Evaluate f(f(2)).

• 4

• -1

• 1

• -2

• 2

Multiple Choice

Evaluate g(g(0)).

• 1

• -5

• 0

• 3

• 2

Drag and Drop

f(x) = ​

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g(x) = ​

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Drag these tiles and drop them in the correct blank above

3x - 5

-x + 3

9x - 20

x

-3x + 8

-3x + 4

Multiple Choice

Given the functions in the image, evaluate g(f(6))

• -10

• -3

• 13

• -14

Multiple Choice

Given the functions in the image, evaluate g(f(1.25))

• 0.25

• 4.25

• -8.75

• -1.25

Multiple Choice

Given the functions in the image, evaluate f(f(⅔))

• -6

• 1

• -4

• -14

Multiple Choice

Given the functions in the image, evaluate g(g(-7))

• 7

• -4

• 11

• -7

Drag and Drop

f(g(x)) = ​

Placeholder

g(f(x)) = ​

Placeholder

f(f(x)) = ​

Placeholder

g(g(x)) = ​

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Drag these tiles and drop them in the correct blank above

-3x + 4

-3x + 8

9x - 20

3x - 5

-x + 3

x

Drag and Drop

Is f(g(x)) = g(f(x))? ​

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.

This means they are ​

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, if they were ​

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, f(x) and g(x) would ​

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.

Drag these tiles and drop them in the correct blank above

No

Yes

not invertible

invertible

the same

different

be inverses

not be inverses

Drag and Drop

Is g(x) an inverse of itself? ​

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.

Is f(x) an inverse of itself? ​

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.

Since g(g(x)) = x, the function ​

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an inverse of itself. Inverses are symmetric with the ​

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; the composition will give an output of x when the functions ​

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.

Drag these tiles and drop them in the correct blank above

Yes

No

is

is not

line, y = x

x-axis

y-axis

are invertible

are not invertible

Composition Formal Definition

The composition of two functions f and g, denoted f∘g, is defined as
(f∘g)(a) = f(g(a)) where the value
of g(a) lies in the domain of f.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

When finished with the above slides….

We will start the stations (part 3) together.

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the wheel as a class and spend a minute or so
discussing our answers.

Part 3: Composing Functions
with Multiple Representations

Drag and Drop

The expression f(2) means the ​

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of a function when the ​

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is 2. In the expression f(2) = 0, the input is ​

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and the output is ​

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, so a coordinate pair representing this expression would be ​

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.

Drag these tiles and drop them in the correct blank above

output

2

0

(2, 0)

(0, 2)

input

Poll

The notes will now only be checked during physical journal checks, so no upload is needed. How do you feel about the content from this lesson?

• I need to look up more examples.

• I am ready to do the stations!

• I will ask my peers and/or the teacher specific questions while doing the stations.

Composition Stations

Travel to six different stations around the classroom. Each station has a handout with a short problem set involving function composition where the functions are expressed in various representations. Students should record their answers on the handout and then check the solutions with each other. Staple all papers together and turn when you are finished.

Station 5 Part a

What are two functions that you observe in h(x)?
One function is “4x − 5” and the other function is
“one divided by some number.”

How could we name each of the functions you saw? f(x) = 4x − 5 and g(x) = 1/x.

In what order should you compose f and g so that the composite function is equivalent to h?
In this case, we should use f as the input for g, so
the composite function is g∘f .