General Mathematics

Quarter 1 – Module 3.1:

Operations on Functions
General Mathematics – Grade 11
Alternative Delivery Mode
Quarter 1 – Module 3.1: Operations on Functions
First Edition, 2020

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General Mathematics
Quarter 1 – Module 3.1 :
Operations on Functions
Introductory Message
For the facilitator:

Welcome to the General Mathematics – Grade 11 Alternative Delivery Mode

(ADM) Module on Functions!

This module was collaboratively designed, developed, and reviewed by

educators both from public and private institutions. Its purpose is to assist you, the

teacher or facilitator in helping the learners meet the standards set by the K to 12

Curriculum while overcoming their personal, social, and economic constraints in

schooling.

This learning resource hopes to engage the learners into guided and

independent learning activities at their own pace and time. Furthermore, this also

aims to help learners acquire the needed 21st century skills while taking into

consideration their needs and circumstances.

In addition to the material in the main text, you will also see this box in the

body of the module:

Notes to the Teacher

This contains helpful tips or strategies that
will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this

module. You also need to keep track of the learners' progress while allowing them to

manage their own learning. Furthermore, you are expected to encourage and assist

the learners as they do the tasks included in the module.

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For the learner:

Welcome to the General Mathematics – Grade 11 Alternative Delivery Mode

(ADM) Module on Operation of Functions!

The operation of functions is a prerequisite of higher mathematics that leads

to a new function with new set of domain and range. This process is done by
combining functions using operations in mathematics.

This module was designed to provide you with fun and meaningful
opportunities for guided and independent learning at your own pace and time. You
will be enabled to process the contents of the learning resource while being an active
learner.

This module has the following parts and corresponding icons:

What I Need to Know This will give you an idea of the skills or
competencies you are expected to learn in the
module.

What I Know This part includes an activity that aims to

check what you already know about the
lesson to take. If you get all the answers
correct (100%), you may decide to skip this
module.

What’s In This is a brief drill or review to help you link

the current lesson with the previous one.

What’s New In this portion, the new lesson will be

introduced to you in various ways such as a
story, a song, a poem, a problem opener, an
activity or a situation.

What is It This section provides a brief discussion of the

lesson. This aims to help you discover and
understand new concepts and skills.

What’s More This comprises activities for independent

practice to solidify your understanding and
skills of the topic. You may check the

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 answers to the exercises using the Answer
Key at the end of the module.

What I Have Learned This includes questions or blank

sentence/paragraph to be filled in to process
what you learned from the lesson.

What I Can Do This section provides an activity which will

help you transfer your new knowledge or skill
into real life situations or concerns.

This is a task which aims to evaluate your

Assessment
level of mastery in achieving the learning
competency.

In this portion, another activity will be given

Additional Activities
to you to enrich your knowledge or skill of the
lesson learned. This also tends retention of
learned concepts.

This contains answers to all activities in the

Answer Key
module.

At the end of this module you will also find:

References This is a list of all sources used in developing

this module.

The following are some reminders in using this module:

1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.

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If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.

We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it!

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 What I Need to Know

At the end of the lesson, the students are expected to perform addition,
subtraction, multiplication, division, and composition of functions M11GM-Ia-3.

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 What I Know

A. Determine what is asked on the presented operation given below.

1. 𝑓(𝑥) = 𝑥 + 4 𝑎𝑛𝑑 𝑔(𝑥) = −2𝑥 − 15; (𝑓 + 𝑔)(𝑥)

2. ℎ(𝑥) = 7𝑥 − 13 𝑎𝑛𝑑 𝑗(𝑥) = 9𝑥 − 16; (ℎ + 𝑗)(𝑥)

3. 𝑘(𝑥) = 𝑥 + 2𝑥 − 8 𝑎𝑛𝑑 𝑚(𝑥) = 7𝑥 + 𝑥 − 2; (𝑘 + 𝑚)(𝑥)

4. 𝑙(𝑥) = 5𝑥 + 12𝑥 − 18 𝑎𝑛𝑑 𝑚(𝑥) = 3𝑥 − 𝑥 − 28; (𝑙 + 𝑚)(𝑥)

5. 𝑛(𝑥) = 𝑥 + 5 𝑎𝑛𝑑 𝑝(𝑥) = 3𝑥 + 15; (𝑛 − 𝑝)(𝑥)

6. 𝑞(𝑥) = 7𝑥 − 12 𝑎𝑛𝑑 𝑟(𝑥) = 3𝑥 − 15; (𝑞 − 𝑟)(𝑥)

7. 𝑎(𝑥) = 7𝑥 − 14𝑥 − 43 𝑎𝑛𝑑 𝑐(𝑥) = 𝑥 + 3𝑥 − 𝑥; (𝑎 − 𝑐)(𝑥)

8. 𝑠(𝑥) = 9𝑥 + 7𝑥 − 13 𝑎𝑛𝑑 𝑡(𝑥) = −2𝑥 + 𝑥 − 8; (𝑠 − 𝑡)(𝑥)

9. 𝑢(𝑥) = 𝑥 + 2 𝑎𝑛𝑑 𝑣(𝑥) = 𝑥 − 2; (𝑢 ∗ 𝑣)(𝑥)

10. 𝐴(𝑥) = 𝑥 − 7 𝑎𝑛𝑑 𝐵(𝑥) = 9𝑥 − 16; (𝐴 ∗ 𝐵)(𝑥)

11. 𝐶(𝑥) = 𝑥 − 1𝑎𝑛𝑑 𝐷(𝑥) = 3𝑥 + 3𝑥 − 21; (𝐶 ∗ 𝐷)(𝑥)

12. 𝐸(𝑥) = 𝑥 + 2𝑥 − 8 𝑎𝑛𝑑 𝐹(𝑥) = 𝑥 − 2; (𝐸 ÷ 𝐹)(𝑥)

13. 𝑓(𝑥) = 𝑥 − 81 𝑎𝑛𝑑 𝑔(𝑥) = −𝑥 + 9; (𝑓 ÷ 𝑔)(𝑥)

14. 𝐺(𝑥) = 12𝑥 + 31𝑥 + 20 𝑎𝑛𝑑 𝐻(𝑥) = 4𝑥 + 5; (𝐺 ÷ 𝐻)(𝑥)

15. 𝑀(𝑥) = 8𝑥 + 22𝑥 + 15 𝑎𝑛𝑑 𝑁(𝑥) = 2𝑥 + 3; (𝑀 ÷ 𝑁)(𝑥)

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 Lesson
Addition and subtraction
1 of Functions

What’s In

Before we proceed with this lesson, let us review addition and subtraction of
polynomials.

A. Add the first (a) to the second B. Subtract the first (a) from the
(b) second (b)

1. (a) 𝑥 + 2 1. (a) 𝑥 + 2

(b) 𝑥 + 1 (b) 𝑥 + 1

2. (a) 𝑥 − 1 2. (a) 𝑥 − 1

(b) 𝑥 + 4 (b) 𝑥 + 4

3. (a) 𝑥 + 2𝑥 + 3 3. (a) 𝑥 + 2𝑥 + 3

(b) 𝑥 + 3𝑥 + 5 (b) 𝑥 + 3𝑥 + 5

4. (a) 2𝑥 − 𝑥 + 1 4. (a) 2𝑥 − 𝑥 + 1

(b) 𝑥 − 2𝑥 − 3 (b) 𝑥 − 2𝑥 − 3

5. (a) 3𝑎 + 2𝑎 − 7 5. (a) 3𝑎 + 2𝑎 − 7

(b) 2𝑎 + 5𝑎 − 11
(b) 2𝑎 + 5𝑎 − 11

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 What’s New

Try to answer the problems presented below.

A. Let us do some addition.

1. If 𝐴 = 𝑥 + 1 and 𝐵 = 𝑥 + 2 what is 𝐴 + 𝐵?

2. If 𝐴 = 4𝑥 + 5 and 𝐵 = 3𝑥 − 4, what is 𝐴 + 𝐵?

3. If 𝑓(𝑥) = 2𝑥 + 3 and ℎ(𝑥) = 𝑥 + 7, what is 𝑓(𝑥) + ℎ(𝑥)?

4. If 𝑃(𝑥) = 𝑥 + 4𝑥 + 8 and 𝐷(𝑥) = 5𝑥 + 3𝑥 + 11, what is 𝑃(𝑥) + 𝐷(𝑥)?

B. Let us try Subtraction this time.

5. If 𝐶 = 3𝑥 + 2 and 𝐷 = 2𝑥 + 1 what is 𝐶 − 𝐷?

6. If 𝑀 = 4𝑥 + 5 and 𝑁 = 3𝑥 − 4, what is 𝑀 − 𝑁?

7. If 𝑓(𝑥) = 2𝑥 + 3 and ℎ(𝑥) = 𝑥 + 7, what is 𝑓(𝑥) − ℎ(𝑥)?

8. If 𝑄(𝑥) = 5𝑥 + 6𝑥 + 18 and 𝑅(𝑥) = 2𝑥 + 3𝑥 − 7, what is 𝑄(𝑥) − 𝑅(𝑥)?

C. Solve using the indicated operation.

9. If 𝑓(𝑥) = 2𝑥 + 3, ℎ(𝑥) = 𝑥 + 7, 𝑃(𝑥) = 5𝑥 + 6𝑥 + 18, and

𝑅(𝑥) = 2𝑥 + 3𝑥 − 7, what is 𝑓(𝑥) + ℎ(𝑥) + 𝑃(𝑥) + 𝑅(𝑥)?

10. If 𝑓(𝑥) = 2𝑥 + 3, ℎ(𝑥) = 𝑥 + 7, 𝑄(𝑥) = 5𝑥 + 6𝑥 + 18, and

𝑅(𝑥) = 2𝑥 + 3𝑥 − 7, what is { 𝑄(𝑥) + 𝑅(𝑥)} − {𝑓(𝑥) + ℎ(𝑥)}?

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 What is It

Let 𝑓 and 𝑔 be functions;

 Their sum, denoted by 𝑓 + 𝑔, is the function denoted by

(𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)

 Their difference, denoted by 𝑓 − 𝑔, is the function denoted by

(𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)

It is best to understand that there is no new rule when adding and subtracting

functions, only new notation. The rules you had been using since elementary algebra

in adding and subtracting expressions is utilized to add and/or subtract functions.

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 What’s More

Operate on the following functions listed below.

 𝑓(𝑥) = 2𝑥 + 5  𝑗(𝑥) = −8𝑥 + 23  𝑝(𝑥) = 𝑥 + 5𝑥 + 13

 𝑔(𝑥) = 3𝑥 − 2  𝑘(𝑥) = 𝑥 − 11𝑥 + 5  𝑞(𝑥) = 𝑥 − 5𝑥 + 17

 ℎ(𝑥) = 12𝑥 − 7  𝑛(𝑥) = 𝑥 − 2𝑥 + 3  𝑟(𝑥) = 9𝑥 − 2𝑥 − 27

A. Addition of Functions

1. (𝑓 + 𝑔)(𝑥) 6. (𝑔 + 𝑛)(𝑥)

2. (ℎ + 𝑗)(𝑥) 7. (𝑟 + 𝑝)(𝑥)

3. (𝑓 + 𝑗)(𝑥) 8. (𝑞 + 𝑝)(𝑥)

4. (𝑔 + ℎ)(𝑥) 9. (𝑛 + 𝑟)(𝑥)

5. (𝑘 + 𝑛)(𝑥) 10. (𝑘 + 𝑟)(𝑥)

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B. Subtraction of Function.

1. (𝑔 − 𝑓)(𝑥) 6. (𝑘 − 𝑛)(𝑥)

2. (𝑓 − 𝑗)(𝑥) 7. (𝑟 − 𝑝)(𝑥)

3. (ℎ − 𝑗)(𝑥) 8. (𝑟 − 𝑞)(𝑥)

4. (ℎ − 𝑓)(𝑥) 9. (𝑝 − 𝑞)(𝑥)

5. (𝑓 − 𝑔)(𝑥) 10. (𝑘 − 𝑗)(𝑥)

C. Try to answer the following.

1. 𝑓(𝑥) + 𝑔(𝑥) − 𝑗(𝑥)

2. (𝑟 + 𝑞)(𝑥) − (𝑝 + 𝑘)(𝑥)

3. (𝑟 − 𝑝)(𝑥) − (𝑛 + 𝑘)(𝑥)

D. Let us try to combine evaluation of function with our topic.

1. (𝑓 + 𝑔)(2)

2. (𝑘 + 𝑛)(−1)

3. (𝑔 − 𝑓)(3)

4. (𝑟 − 𝑞)(2)

5. (𝑘 − 𝑗)(𝑎 + 1)

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 What I Have Learned

Fill in the blanks.

There is NO NEW RULE in addition and subtraction of functions only new

notation.

Thus, if 𝑓 and 𝑔 are functions:

To add function (𝑓 + 𝑔)(𝑥) = ________________________________; and

To subtract function (𝑓 − 𝑔)(𝑥) = ___________________________

in function notation.

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 What I Can Do

Solve the presented problems with the aid of addition and subtraction of function.

1. A 15-meter wire is to be cut into two

parts with different lengths. If one part
(𝑥) will be used to form a square and
the other part a circle as illustrated,
represent the total area 𝐴(𝑥) that
enclosed the two figures.

To solve the problem, follow these steps:

a. Create a function model that will represent the Area of the square 𝑠(𝑥).
Hint: Recall the characteristics of the parts of a square as well as its
formulas.

b. Create a function model that represents the area of the circle 𝑐(𝑥). Hint:
Recall the characteristics of the parts of a circle as well as its formulas.

c. The total area that enclosed these figures is: 𝐴(𝑥) = 𝑠(𝑥) + 𝑐(𝑥) or simply
𝐴(𝑥) = (𝑠 + 𝑐)(𝑥). 𝑆tate your answer in more comprehensible function
model.

2. In an Economics subject, the profit 𝑃(𝑥) in a certain business can be expressed

by subtracting the cost 𝐶(𝑥) from revenue 𝑅(𝑥) or in function model as 𝑃(𝑥) =
𝑅(𝑥) − 𝐶(𝑥). If the revue is expressed as 𝑅(𝑥) = 13𝑥 + 12𝑥 + 1 and the
cost 𝐶(𝑥) = 8𝑥 − 5𝑥 − 23, how will you express the profit gained by the certain
business?

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3. A Marine plywood with an area of
𝑝(𝑥) = 11𝑥 − 5𝑥 − 31 is to be used to make a
divider and a small footrest. What is the area
left for the footrest if the area of the wood
used in the divider is 𝑞(𝑥) = 3𝑥 − 7𝑥 + 9?

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 Assessment

Operate on the following functions listed below.

 𝑓(𝑥) = 3𝑥 + 4  𝑗(𝑥) = −3𝑥 + 33  𝑝(𝑥) = 3𝑥 + 8𝑥 + 11

 𝑔(𝑥) = 5𝑥 − 7  𝑘(𝑥) = 𝑥 − 7𝑥 + 21  𝑞(𝑥) = 7𝑥 − 3𝑥 + 21

 ℎ(𝑥) = 25𝑥 − 11  𝑛(𝑥) = 2𝑥 − 2𝑥 + 3  𝑟(𝑥) = 9𝑥 − 3𝑥 − 17

1. (𝑓 + 𝑔)(𝑥)

2. (ℎ + 𝑗)(𝑥)

3. (𝑞 + 𝑝)(𝑥)

4. (𝑔 + 𝑛)(𝑥)

5. (𝑘 + 𝑟)(𝑥)

6. (𝑓 − 𝑗)(𝑥)

7. (ℎ − 𝑓)(𝑥)

8. (𝑓 − 𝑔)(𝑥)

9. (𝑟 − 𝑞)(𝑥)

10. (𝑟 − 𝑝)(𝑥)

11. (𝑓 + 𝑔)(𝑥) − (𝑗)(𝑥)

12. (𝑟 − 𝑝)(𝑥) − (𝑛 + 𝑘)(𝑥)

13. (𝑘 + 𝑛)(−5)

14. (𝑟 − 𝑞)(2𝑎)

15. (𝑘 − 𝑗)(3𝑎 + 2)

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 Additional Activities

Let us take the challenge! Answer the problems below using the given functions.

 𝑔(𝑥) = 4𝑥 − 11  ℎ(𝑥) = 3𝑥 + 1  𝑗(𝑥) = 2𝑥 − 4𝑥 + 1

1. Express the function 𝑓(𝑥) = 3𝑥 + 4𝑥 − 10 as a sum or difference of the

functions above.

2. Express the function 𝑓(𝑥) = 2𝑥 − 10 as a sum or difference of the functions

above.

3. Express the function 𝑓(𝑥) = 5𝑥 − 4𝑥 + 2 as a sum or difference of the

functions above.

4. Express the function 𝑓(𝑥) = 𝑥 + 4𝑥 as a sum or difference of the functions

above.

5. Express the function 𝑓(𝑥) = −2𝑥 + 8𝑥 − 12 as a sum or difference of the

functions above.

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 Lesson
Multiplication and Division
2 of Functions

What’s In

Before we proceed with this lesson, let us review multiplication and division of
polynomials.

A. Multiply the first (a) by the B. Divide the first (a) by the
second (b) second (b)

1. (a) 𝑥 + 1 1. (a) 𝑥 − 4

(b) 𝑥 + 3 (b) 𝑥 + 2

2. (a) 𝑥 − 2 2. (a) 𝑥 + 3𝑥 − 4

(b) 𝑥 + 5 (b) 𝑥 + 4

3. (a) 𝑥 + 𝑥 + 1 3. (a) 3𝑥 − 2𝑥 − 8

(b) 𝑥 + 3𝑥 + 2 (b) 3𝑥 + 4

4. (a) 3𝑥 − 3𝑥 + 2 4. (a) 𝑥 − 𝑥 − 5𝑥 − 3

(b) 2𝑥 − 3𝑥 − 7 (b) 𝑥 − 2𝑥 − 3

5. (a) 3𝑎 + 3𝑎 − 17 5. (a) 3𝑎 + 63𝑎 − 47𝑎 − 168𝑎 + 104

(b) 5𝑎 + 2𝑎 − 1 (b) 𝑎 + 21𝑎 − 13

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 What’s New

Try to answer the problems presented below.

A. Let us do some Multiplication.

1. If 𝐴 = 𝑥 + 2 and 𝐵 = 𝑥 + 3 what is 𝐴 ∗ 𝐵?

2. If 𝐴 = 7𝑥 + 6 and 𝐵 = 2𝑥 − 3, what is 𝐴 ∗ 𝐵?

3. If 𝑓(𝑥) = 3𝑥 + 2 and ℎ(𝑥) = 𝑥 + 9, what is 𝑓(𝑥) ∗ ℎ(𝑥)?

4. If 𝑃(𝑥) = 2𝑥 + 4 and 𝐷(𝑥) = 7𝑥 + 2𝑥 + 13, what is 𝑃(𝑥) ∗ 𝐷(𝑥)?

5. If 𝑘(𝑥) = 7𝑥 + 5𝑥 and 𝑡(𝑥) = 𝑥 + 𝑥 − 21, what is 𝑘(𝑥) ∗ 𝑡(𝑥)?

B. Let us try Division this time.

6. If 𝐶 = 4𝑥 − 4𝑥 + 1 and 𝐷 = 2𝑥 − 1 what is 𝐶 ÷ 𝐷?

7. If 𝑀 = 12𝑥 − 17𝑥 − 40 and 𝑁 = 3𝑥 − 8, what is 𝑀 ÷ 𝑁?

8. If 𝑓(𝑥) = 6𝑥 + 31𝑥 + 33 and ℎ(𝑥) = 3𝑥 + 11, what is 𝑓(𝑥) ÷ ℎ(𝑥)?

9. If 𝑄(𝑥) = 10𝑥 − 2𝑥 − 29𝑥 + 9𝑥 − 72 and 𝑅(𝑥) = 2𝑥 − 9, what is

𝑄(𝑥) ÷ 𝑅(𝑥)?

10. If 𝑣(𝑥) = 5𝑥 + 15𝑥 + 18𝑥 + 54𝑥 and 𝑤(𝑥) = 𝑥 + 3𝑥,

what is 𝑣(𝑥) ÷ 𝑤(𝑥)?

18
 What is It

Let 𝑓 and 𝑔 be functions;

 Their product, denoted by 𝑓 ∗ 𝑔, is the function denoted by

(𝑓 ∗ 𝑔)(𝑥) = 𝑓(𝑥) ∗ 𝑔(𝑥)

 Their quotient, denoted by 𝑓 ÷ 𝑔, is the function denoted by

(𝑓 ÷ 𝑔)(𝑥) = 𝑓(𝑥) ÷ 𝑔(𝑥)

It is best to understand that there is no new rule in Multiplying and Dividing functions,
only new notation. The rules you had been using since elementary algebra in
multiplying and dividing expressions is utilized to multiply and/or divide functions.

19
 What’s More

From the function presented, determine the following function.

A. Multiplication of Function

1. 𝑓(𝑥) = 𝑥 + 2 𝑎𝑛𝑑 𝑔(𝑥) = 3𝑥 − 5; (𝑓 ∗ 𝑔)(𝑥)

2. ℎ(𝑥) = 7𝑥 − 13 𝑎𝑛𝑑 𝑗(𝑥) = −3𝑥 + 15; (ℎ ∗ 𝑗)(𝑥)

3. 𝑓(𝑥) = 𝑥 + 2 𝑎𝑛𝑑 𝑗(𝑥) = −3𝑥 + 15; (𝑓 ∗ 𝑗)(𝑥)

4. 𝑔(𝑥) = 3𝑥 − 5 𝑎𝑛𝑑 ℎ(𝑥) = 7𝑥 − 13; (𝑔 ∗ ℎ)(𝑥)

5. 𝑘(𝑥) = 9𝑥 − 16 𝑎𝑛𝑑 𝑛(𝑥) = 3𝑥 + 𝑥 − 10; (𝑘 ∗ 𝑛)(𝑥)

6. 𝑔(𝑥) = 3𝑥 − 5 𝑎𝑛𝑑 𝑛(𝑥) = 3𝑥 + 𝑥 − 10; (𝑔 ∗ 𝑛)(𝑥)

7. 𝑟(𝑥) = 5𝑥 − 2𝑥 − 27 𝑎𝑛𝑑 𝑓(𝑥) = 𝑥 + 2; (𝑟 ∗ 𝑓)(𝑥)

8. 𝑞(𝑥) = 𝑥 − 5𝑥 + 17 𝑎𝑛𝑑 𝑔(𝑥) = 3𝑥 − 5; (𝑞 ∗ 𝑔)(𝑥)

9. 𝑗(𝑥) = −3𝑥 + 15 𝑎𝑛𝑑 𝑟(𝑥) = 5𝑥 − 2𝑥 − 27; (𝑗 ∗ 𝑟)(𝑥)

10. 𝑓(𝑥) = 𝑥 + 2 𝑎𝑛𝑑 𝑝(𝑥) = 4𝑥 − 19𝑥 + 9𝑥 − 2; (𝑓 ∗ 𝑝)(𝑥)

20
B. Division of Function.

a. 𝑓(𝑥) = 𝑥 + 3𝑥 + 2 𝑎𝑛𝑑 𝑠(𝑥) = 𝑥 + 1; (𝑓 ÷ 𝑠)(𝑥)

b. 𝑔(𝑥) = 𝑥 + 6𝑥 + 8 𝑎𝑛𝑑 𝑡(𝑥) = 𝑥 + 2;(𝑓 ÷ 𝑡)(𝑥)

c. ℎ(𝑥) = 𝑥 − 4 𝑎𝑛𝑑 𝑢(𝑥) = 𝑥 − 2;(ℎ ÷ 𝑢)(𝑥)

d. 𝑗(𝑥) = 𝑥 + 2𝑥 − 8 𝑎𝑛𝑑 𝑣(𝑥) = 𝑥 + 4;(𝑗 ÷ 𝑣)(𝑥)

e. 𝑘(𝑥) = 12𝑥 + 31𝑥 + 20 𝑎𝑛𝑑 𝑤(𝑥) = 4𝑥 + 5;(𝑘 ÷ 𝑤)(𝑥)

f. 𝑚(𝑥) = 8𝑥 + 22𝑥 + 15 𝑎𝑛𝑑 𝐹(𝑥) = 2𝑥 + 3;(𝑚 ÷ 𝐹)(𝑥)

g. 𝑛(𝑥) = −32𝑥 + 52𝑥 + 115 𝑎𝑛𝑑 𝐺(𝑥) = 4𝑥 + 5;(𝑛 ÷ 𝐺)(𝑥)

h. 𝑝(𝑥) = −96𝑥 + 332𝑥 − 161 𝑎𝑛𝑑 𝑅(𝑥) = −8𝑥 + 23;(𝑝 ÷ 𝑅)(𝑥)

i. 𝑞(𝑥) = 12𝑥 + 29𝑥 − 69𝑥 + 28 𝑎𝑛𝑑 𝐶(𝑥) = 12𝑥 − 7;(𝑞 ÷ 𝐶)(𝑥)

j. 𝑟(𝑥) = 3𝑥 + 7𝑥 − 26𝑥 − 16𝑥 + 32 𝑎𝑛𝑑 𝐷(𝑥) = 𝑥 + 3𝑥 − 4;(𝑟 ÷ 𝐷)(𝑥)

21
 What I Have Learned

Fill in the blanks.

There is NO NEW RULE in multiplication and division of functions only new

notation.

Thus, if 𝑓 and 𝑔 are functions:

To multiply function (𝑓 ∗ 𝑔)(𝑥) = ________________________________; and

To divide function (𝑓 ÷ 𝑔)(𝑥) = ___________________________

in function notation.

22
 What I Can Do

Real-life problems can be represented using function models and solved

mathematically. Solve the presented problem using Multiplication/or Division of
functions.

The velocity of the car is expressed

as 𝑉(𝑥) = 4𝑥 + 3 − 3 and its time
as 𝑡(𝑥) = 2𝑥 + 5. Create a function
model about the distance travelled
of the car.

To solve the problem, follow these steps:

a. State the function model representing the velocity of the car.

b. State the function model expressing the time lapse of the car.

c. Recall the formula in finding the velocity of a moving object with respect
to distance and time. Derived a formula of distance from that formula.

d. Create the function model for the distance travelled by the car. Simply
state your answer in a more comprehensible function model.

23
 Assessment

From the function presented, determine the following functions.

1. 𝑓(𝑥) = 2𝑥 𝑎𝑛𝑑 𝑔(𝑥) = 7𝑥 − 15; (𝑓 ∗ 𝑔)(𝑥)

2. ℎ(𝑥) = 7𝑥 − 1 𝑎𝑛𝑑 𝑗(𝑥) = −3𝑥 + 5; (ℎ ∗ 𝑗)(𝑥)

3. 𝑔(𝑥) = 3𝑥 − 1 𝑎𝑛𝑑 ℎ(𝑥) = 3𝑥 − 11; (𝑔 ∗ ℎ)(𝑥)

4. 𝑘(𝑥) = 9𝑥 − 16 𝑎𝑛𝑑 𝑛(𝑥) = 3𝑥 − 10; (𝑘 ∗ 𝑛)(𝑥)

5. 𝑔(𝑥) = 𝑥 − 1 𝑎𝑛𝑑 𝑛(𝑥) = 𝑥 + 𝑥 − 1; (𝑔 ∗ 𝑛)(𝑥)

6. 𝑞(𝑥) = 2𝑥 − 3𝑥 + 7 𝑎𝑛𝑑 𝑔(𝑥) = 4𝑥 − 5; (𝑞 ∗ 𝑔)(𝑥)

7. 𝑗(𝑥) = −3𝑥 + 14 𝑎𝑛𝑑 𝑟(𝑥) = 3𝑥 − 2𝑥 − 2; (𝑗 ∗ 𝑟)(𝑥)

8. 𝑓(𝑥) = 7𝑥 + 2 𝑎𝑛𝑑 𝑝(𝑥) = 𝑥 − 9𝑥 + 𝑥 − 1; (𝑓 ∗ 𝑝)(𝑥)

9. 𝑓(𝑥) = 6𝑥 + 15𝑥 + 5𝑥 − 6 𝑎𝑛𝑑 𝑠(𝑥) = 2𝑥 + 3; (𝑓 ÷ 𝑠)(𝑥)

10. 𝑔(𝑥) = 3𝑥 − 𝑥 − 35𝑥 − 22 𝑎𝑛𝑑 𝑡(𝑥) = 3𝑥 + 2;(𝑓 ÷ 𝑡)(𝑥)

11. ℎ(𝑥) = 2𝑥 − 22𝑥 + 4𝑥 − 44 𝑎𝑛𝑑 𝑢(𝑥) = 𝑥 − 11;(ℎ ÷ 𝑢)(𝑥)

12. 𝑗(𝑥) = −16𝑥 −32𝑥 + 46𝑥 + 92 𝑎𝑛𝑑 𝑣(𝑥) = 2𝑥 + 4;(𝑗 ÷ 𝑣)(𝑥)

13. 𝑛(𝑥) = 48𝑥 +180𝑥 − 28𝑥 − 105 𝑎𝑛𝑑 𝐺(𝑥) = 4𝑥 + 15;(𝑛 ÷ 𝐺)(𝑥)

14. 𝑝(𝑥) = −2𝑥 +25𝑥 + 101𝑥 − 124 𝑎𝑛𝑑 𝑅(𝑥) = −2𝑥 + 31;(𝑝 ÷ 𝑅)(𝑥)

15. 𝑟(𝑥) = 6𝑥 − 𝑥 − 36𝑥 + 4𝑥 + 48 𝑎𝑛𝑑 𝐷(𝑥) = 2𝑥 + 𝑥 − 6;(𝑟 ÷ 𝐷)(𝑥)

24
 Additional Activities

Let us take the challenge! Answer the problems below using the given functions.

 𝑔(𝑥) = 2𝑥 + 1  ℎ(𝑥) = 𝑥 + 11  𝑗(𝑥) = 2𝑥 + 𝑥 + 22𝑥 +

11

1. Express the function 𝑓(𝑥) = 4𝑥 + 4𝑥 + 25𝑥 + 44𝑥 + 11 as a product or

quotient of the functions above.

2. Express the function 𝑓(𝑥) = 2𝑥 + 𝑥 + 22𝑥 + 11 as a product or quotient of

the functions above.

3. Express the function 𝑓(𝑥) = 2𝑥 + 𝑥 + 44𝑥 + 22𝑥 + 242𝑥 + 121 as a

product or quotient of the functions above.

4. Express the function 𝑓(𝑥) = 2𝑥 + 1 as a product or quotient of the functions

above.

5. Express the function 𝑓(𝑥) = 𝑥 + 11 as a product or quotient of the

functions above.

25
 26
What’s More: Lesson 1
A.1. 5𝑥 + 3
What’s More: Lesson 1 A.2. 4𝑥 + 16
A.3. −6𝑥 + 28
B.6. −9𝑥 + 2 A.4. 15𝑥 − 9
B.7. 8𝑥 − 7𝑥 − 40 A.5. 2𝑥 − 13𝑥 + 8
B.8. 8𝑥 + 3𝑥 − 44 A.6. 𝑥 +𝑥+1
What I Can Do: Lesson 1 B.9. 10𝑥 − 4 A.7. 10𝑥 + 3𝑥 − 14
B.10. 𝑥 − 3𝑥 − 18 A.8. 2𝑥 + 30
1.a. 𝑠(𝑥) = 𝑥 A.9. 10𝑥 − 4𝑥 − 24
( )
A.10. 10𝑥 − 13𝑥 − 22
1.b. 𝑐(𝑥) = C.1. 13𝑥 − 20
( ) C.2. 8𝑥 − 𝑥 − 28
1.c. 𝐴(𝑥) = 𝑥 + B.1. 𝑥−7
C.3. 6𝑥 + 6𝑥 − 48
B.2. 10𝑥 − 18
B.3. 20𝑥 − 30
2. 𝑃(𝑥) = 5𝑥 + 17𝑥 + 24 D.1. 13 B.4. 10𝑥 − 12
D.2. 23 B.5. −𝑥 + 7
3. 𝐴(𝑥) = 8𝑥 + 2𝑥 − 40
D.3. -4 B.6. −9𝑥 + 2
D.4. -6 B.7. 8𝑥 − 7𝑥 − 40
D.5. 𝑎 − 𝑎 − 20 B.8. 8𝑥 + 3𝑥 − 44
B.9. 10𝑥 − 4
B.10. 𝑥 − 3𝑥 − 18
13𝑥 − 20
What I Know:
What’s New: Lesson 1 What’s In: Lesson 1 1. −𝑥 − 9
2. 9𝑥 + 7𝑥 − 29
1. 2𝑥 + 3 3. 8𝑥 + 3𝑥 − 10
1. 2𝑥 + 3 2. 2𝑥 + 3 4. 8𝑥 + 11𝑥 − 46
2. 7𝑥 + 1 3. 2𝑥 + 5𝑥 + 8 5. −2𝑥 − 10
3. 3𝑥 + 10 4. 3𝑥 − 3𝑥 − 2 6. 4𝑥 + 3
4. 6𝑥 + 7𝑥 + 19 5. 5𝑎 + 7𝑎 − 18 7. −𝑥 + 4𝑥 − 13𝑥 − 43
5. 𝑥+1
8. 11𝑥 + 6𝑥 − 5
6. 𝑥+9
9. 𝑥 −4
7. 𝑥−4 1. -1 10. 9𝑥 − 79𝑥 + 112
8. 3𝑥 + 3𝑥 + 25 2. 5 11. 3𝑥 + 3𝑥 − 24𝑥 −
9. 7𝑥 + 12𝑥 + 21 3. 𝑥+2 3𝑥 + 21
10. 7𝑥 + 6𝑥 + 1 4. −𝑥 − 𝑥 − 4 12.𝑥+4
5. −𝑎 + 3𝑎 − 4 13.−𝑥 − 9
14.3𝑥 + 4
15.4𝑥 + 5
Answer Key
 27
Assessment: Lesson 2
1. 14𝑥 − 30𝑥 What I Can Do:
2. −21𝑥 + 38𝑥 − 5 Lesson 2
3. 9𝑥 − 36𝑥 + 11
Additional
4. 27𝑥 − 138𝑥 + 160 a. 𝑉(𝑥) = 4𝑥 + 3 − 3
Activities:
5. 𝑥 − 2𝑥 + 1
Lesson 2
6. 8𝑥 − 22𝑥 + 43𝑥 − 35 b. 𝑡(𝑥) = 2𝑥 + 5
1. 𝑔(𝑥) ∗ 𝑗(𝑥) 7. −9𝑥 + 48𝑥 − 22𝑥 −
2. 𝑔(𝑥) ∗ ℎ(𝑥) 28
c. 𝑉= 𝑡ℎ𝑢𝑠 𝑑 = 𝑉𝑡
3. ℎ(𝑥) ∗ 𝑗(𝑥) 8. 7𝑥 − 61𝑥 − 11𝑥 − 5
4. 𝑗(𝑥) ÷ ℎ(𝑥) 9. 𝑥−2
d. 𝑑(𝑥) = 𝑉(𝑥) ∗ 𝑡(𝑥)
5. 𝑗(𝑥) ÷ 𝑔(𝑥) 10. 3𝑥 + 3𝑥 − 2
11. 𝑥 − 𝑥 − 11
𝑑(𝑥) = 8𝑥 + 202𝑥
12. 2𝑥 + 4
13. −8𝑥 + 23
14. 12𝑥 − 7
15. 𝑥 + 3𝑥 − 4
16. 3𝑥 − 2𝑥 − 8
What’s More: Lesson 2 What’s New: Lesson 2
What’s More:
Lesson 2 A. A.1. 𝑥 + 12𝑥 + 6
A.2. 14𝑥 − 9𝑥 − 18
B. 1. 3𝑥 + 𝑥 − 10
A.3. 3𝑥 + 29𝑥 + 18
2. −21𝑥 + 144𝑥 − 195
1. 𝑥+2 A.4. 14𝑥 + 4𝑥 + 54𝑥 +
3. −3𝑥 + 9𝑥 + 30
2. 𝑥+4 8𝑥 + 52
4. 21𝑥 − 74𝑥 + 65
3. 𝑥+2 A.5. 7𝑥 + 12𝑥 − 142𝑥 −
5. 27𝑥 + 9𝑥 − 138𝑥 − 16𝑥 +
4. 𝑥−2 105𝑥
160
5. 3𝑥 + 4 6. 9𝑥 − 12𝑥 − 35𝑥 + 50
6. 4𝑥 + 5 B.1.2𝑥 − 1
7. 5𝑥 + 8𝑥 − 31𝑥 − 54
7. −8𝑥 + 23 B.2.4𝑥 + 5
8. 3𝑥 − 20𝑥 + 76𝑥 − 85 B.3.2𝑥 + 3
8. 12𝑥 − 7 9. −15𝑥 + 81𝑥 + 519𝑥 − 405
9. 𝑥 + 3𝑥 − 4 B.4.5𝑥 − 𝑥 + 8
10. 4𝑥 − 11𝑥 − 29𝑥 + 16𝑥 − 4
10. 3𝑥 − 2𝑥 − 8 B.5.5𝑥 + 18
What’s In: Lesson 2 Assessment: Lesson 1
1. 𝑥 + 4𝑥 + 3 1. 8𝑥 − 3
2. 𝑥 + 3𝑥 − 10 2. 22𝑥 + 22
3. 𝑥 + 4𝑥 + 6𝑥 + 5𝑥 + 2 3. 10𝑥 + 5𝑥 + 32
12𝑥 − 21𝑥 − 4𝑥 + Additional
4. 4. 2𝑥 + 3𝑥 − 4
15𝑥 − 14 Activities: Lesson 1 5. 10𝑥 − 10𝑥 + 4
5. 15𝑎 + 21𝑎 − 82𝑎 − 6. 6𝑥 − 29
1. 𝑔(𝑥) + ℎ(𝑥)
37𝑎 + 17 7. 22𝑥 − 15
2. 𝑔(𝑥) + 𝑗(𝑥)
8. −2𝑥 + 11
3. 𝑗(𝑥) + ℎ(𝑥)
1. 𝑥−2 9. 2𝑥 − 38
4. ℎ(𝑥) − 𝑗(𝑥)
2. 𝑥−1 10. 6𝑥 − 11𝑥 − 28
5. 𝑔(𝑥) − 𝑗(𝑥)
3. 𝑥−2 11. 11𝑥 − 36
4. 𝑥+1 12. 3𝑥 − 2𝑥 − 52
5. 3𝑎 − 8 13. 144
14. 8𝑎 − 38
15. 9𝑎 − 16
References
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Mathematics. Diliman, Quezon City. 2016.

Department of Education. General Mathematics Learner’s Manual. Pasig City. 2016.

Department of Education. K to 12 Most Essential Learning Competencies with

Corresponding CG Codes. Pasig City. 2020.

Fernando B. Orines. Next Century Mathematics General Mathematics. Quezon City.

Phoenix Publishing House. 2016.

Margaret L. Lial and Charles D. Miller. Mathematics: with application in the

Management, Natural and Social Studies.Illinios. Scott, Foresman and
Company. 1974.

Max A. Sobel and Norbert Lerner. Algebra and Trigonometry: a pre-calculus Approach.
New Jersey. Prentice-Hall, Inc. 1983.

Raymond A Barnett and Michael R, Ziegler. Precalculus Functions and graphs. New
York. McGraw Hill, Inc. 1993.

Catalina Dinio Mijares.2004. College Algebra Revised Edition. Mandaluyong City.

National Book Store. 2004.

28
For inquiries or feedback, please write or call:

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Schools Division of Bataan - Curriculum Implementation Division
Learning Resources Management and Development Section (LRMDS)

Provincial Capitol Compound, Balanga City, Bataan

Telefax: (047) 237-2102

Email Address: bataan@deped.gov.ph