CO_Q1_General Mathematic SHS Module 3

General Mathematics Quarter 1 – Module 3:

Operations on Functions

General Mathematics Alternative Delivery Mode Quarter 1 – Module 3: Operations on Functions First Edition, 2021 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

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Development Team of the Module

Writers: Nestor N. Sandoval Editors: Elizabeth B. Dizon, Anicia J. Villaruel, Roy O. Natividad Reviewers: Fritz A. Caturay, Necitas F. Constante, Celestina M. Alba, Jerome A. Chavez, Edna G. Adel, Lirio G. Parale, Flora P. Segovia Illustrator: Dianne C. Jupiter Layout Artist: Noel Rey T. Estuita, Mark Anthony N. Banta Management Team: Francis Cesar B. Bringas

Job S. Zape Jr. Ramonito Elumbaring Reicon C. Condes Elaine T. Balaogan Fe M. Ong-ongowan Elias A. Alicaya Jr. Gregorio A. Co Jr. Gregorio T. Mueco Herbert D. Perez Lorena S. Walangsumbat Jee-Ann O. Borines Asuncion C. Ilao

General Mathematic Quarter 1 – Module 3:

Operations on Functions

Introductory Message This Self-Learning Module (SLM) is prepared so that you, our dear learners,

can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

1 CO_Q1_General Mathematic SHS Module 3

What I Need to Know

In this module, the different operations on functions were discussed. Examples were provided for you to be able to learn the five (5) operations: addition, subtraction, multiplication, division and composition of functions. Aside from algebraic solutions, these examples were illustrated, represented in tables and/or mapping diagram for better understanding of the concepts. Activities were provided to enhance your learning. Finally, your task is to answer a guided real-world problem that involves operations on functions.

After going through this module, you are expected to:

1. define operations on functions; 2. identify the different operations on functions; and 3. perform addition, subtraction, multiplication, division, and composition of functions.

What I Know

Write the letter of the correct answer on a separate sheet of paper.

1. The statement "𝑝𝑝(𝑥𝑥) − 𝑞𝑞(𝑥𝑥) is the same as 𝑞𝑞(𝑥𝑥) − 𝑝𝑝(𝑥𝑥)", 𝑝𝑝(𝑥𝑥) ≠ 𝑞𝑞(𝑥𝑥) is _____. a. always true b. never true c. sometimes true d. invalid

2. Given ℎ(𝑥𝑥) = 2𝑥𝑥2 − 7𝑥𝑥 and 𝑟𝑟(𝑥𝑥) = 𝑥𝑥2 + 𝑥𝑥 − 1, find (ℎ + 𝑟𝑟)(𝑥𝑥).

a. 2𝑥𝑥2 – 1 b. 3𝑥𝑥2 + 6𝑥𝑥 – 1 c. 3𝑥𝑥4 − 6𝑥𝑥2 – 1 d. 3𝑥𝑥2 − 6𝑥𝑥 – 1

3. Given: 𝑓𝑓(𝑎𝑎) = 2𝑎𝑎 + 1 and 𝑔𝑔(𝑎𝑎) = 3𝑎𝑎 − 3. Find 𝑓𝑓(𝑎𝑎) + 𝑔𝑔(𝑎𝑎) 𝑎𝑎. 5𝑎𝑎 − 2 b. −5𝑎𝑎 + 2 c. −2𝑎𝑎 + 1 d. −6𝑎𝑎 − 1

4. 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 − 4 and ℎ(𝑥𝑥) = 2𝑥𝑥 − 7 Find (𝑔𝑔 + ℎ)(3). a. -7 b. 1 c.-1 d. 8

5. 𝑓𝑓(𝑥𝑥) = 6𝑥𝑥2 + 7𝑥𝑥 + 2 and 𝑔𝑔(𝑥𝑥) = 5𝑥𝑥2 − 𝑥𝑥 − 1, find (𝑓𝑓 − 𝑔𝑔)(𝑥𝑥).

a. 𝑥𝑥2 + 8𝑥𝑥 + 3 b. 5𝑥𝑥2 + 8𝑥𝑥 – 1 c. 𝑥𝑥2 + 6𝑥𝑥 – 1 d. 𝑥𝑥2 + 8𝑥𝑥 − 1

6. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 − 8 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 3, Find 𝑓𝑓(𝑥𝑥) • 𝑔𝑔(𝑥𝑥) a. 𝑥𝑥2 + 24 b. 𝑥𝑥2 − 5𝑥𝑥 + 24 c. 𝑥𝑥2 − 5𝑥𝑥 − 24 d. 𝑥𝑥2 + 5𝑥𝑥 + 24

2 CO_Q1_General Mathematic SHS Module 3

7. If 𝑝𝑝(𝑥𝑥) = 𝑥𝑥 − 1 and 𝑞𝑞(𝑥𝑥) = 𝑥𝑥 − 1, what is 𝑝𝑝(𝑥𝑥) • 𝑞𝑞(𝑥𝑥) a. 𝑥𝑥2 + 1 b. 𝑥𝑥2 + 2𝑥𝑥 − 1 c. 𝑥𝑥2 − 2𝑥𝑥 + 1 d. 𝑥𝑥2 − 1

8. Given ℎ(𝑥𝑥) = 𝑥𝑥 − 6 𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠(𝑥𝑥) = 𝑥𝑥2 − 13𝑥𝑥 + 42. Find ℎ

𝑠𝑠(𝑥𝑥).

a. 1𝑥𝑥−7

b. 𝑥𝑥 − 7 c. 𝑥𝑥−6𝑥𝑥−7

d. 𝑥𝑥 − 6

9. 𝑔𝑔(𝑥𝑥) = 6𝑥𝑥 − 7 and ℎ(𝑥𝑥) = 5𝑥𝑥 − 1, Find 𝑔𝑔(ℎ(𝑥𝑥)) a. −9𝑥𝑥 + 11 b. 9𝑥𝑥2 + 4𝑥𝑥 c.30𝑥𝑥 + 13 d. 30𝑥𝑥 − 13

10.

If 𝑗𝑗(𝑥𝑥) = √𝑥𝑥 + 6 and 𝑘𝑘(𝑥𝑥) = 9 − 𝑥𝑥. Find 𝑗𝑗(𝑘𝑘(−1))

a. 9 − √5 b. √14 c. 16 d. 4

For numbers 11-13, refer to figure below

11. Evaluate 𝑝𝑝(5)

a. 0 b. 3 c. 2 d. 7

12. Find 𝑞𝑞(𝑝𝑝(0)) a. -3 b. 1 c. -3 d. -5

13. Find (𝑞𝑞 ∘ 𝑝𝑝)(3)

a. 3 b. 5 c. 7 d. -1

For numbers 14-15, refer to the table of values below

𝑚𝑚(𝑥𝑥) = 3𝑥𝑥 − 5 𝑎𝑎(𝑥𝑥) = 𝑥𝑥2 − 2𝑥𝑥 + 1 𝑥𝑥 0 1 2 3 4 5 6 7 8

𝑚𝑚(𝑥𝑥) -5 -2 1 4 7 10 13 16 19 𝑎𝑎(𝑥𝑥) 1 0 1 4 9 16 25 36 49

14. Find 𝑚𝑚𝑛𝑛

(7)

a. 49 b. 9

4 c. 1 d. 0

15. Find (𝑎𝑎 ∘ 𝑚𝑚)(4)

a. 9 b. 16 c. 19 d. 36

3 CO_Q1_General Mathematic SHS Module 3

Lesson

1 Operations on Functions

Operations on functions are similar to operations on numbers. Adding, subtracting and multiplying two or more functions together will result in another function. Dividing two functions together will also result in another function if the denominator or divisor is not the zero function. Lastly, composing two or more functions will also produce another function.

The following are prerequisite skills before moving through this module:

Rules for adding, subtracting, multiplying and dividing fractions and algebraic expressions, real numbers (especially fractions and integers).

Evaluating a function.

A short activity was provided here for you to help in recalling these competencies. If you feel that you are able to perform those, you may skip the activity below. Enjoy!

What’s In

SECRET MESSAGE Answer each question by matching column A with column B. Write the letter of the correct answer at the blank before each number. Decode the secret message below using the letters of the answers.

Column A Column B _____1. Find the LCD of 1

3 and 2

7. A. (x + 4)(x− 3)

_____2. Find the LCD of 3x−2

and 1x+3

C. 4x+7x2+x−6

_____3. Find the sum of 13

and 27. D. (𝑥𝑥−3)(𝑥𝑥+5)

(x−6)(x+3)

_____4. Find the sum of 2x

+ 5x E. (𝑥𝑥 − 2)(𝑥𝑥 + 3) or x2 + x − 6

_____5. Find the product of 38

and 125

. G.𝑥𝑥+4x+2

_____6. Find the sum of 3x−2

and 1x+3

H. (x + 1)(x− 6) For numbers 7-14, find the factors.

_____7. x2 + x − 12 I. 1321

_____8. x2 − 5x − 6 L. (𝑥𝑥 − 4(𝑥𝑥 − 3)

4 CO_Q1_General Mathematic SHS Module 3

_____9. x2 + 6x + 5 M. −5 _____10. x2 + 7x + 12 N. 21 _____11. x2 − 7x + 12 O. (𝑥𝑥 − 5)(𝑥𝑥 − 3) _____12. x2 − 5x − 14 R. (x + 4)(x + 3) _____13. x2 − 8x + 15 S. (𝑥𝑥 − 7)(𝑥𝑥 − 5) _____14. x2 − 12x + 35 T. 9

10

_____15. Find the product of x2+x−12x2−5x−6

and x2+6x+5

x2+7x+12. U. (𝑥𝑥 − 7(𝑥𝑥 + 2)

_____16. Divide x2+x−12

x2−5x−14by x2−8x+15

x2−12x+35 W. 7

𝑥𝑥

_____17. In the function f(x) = 4 − x2, 𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎 𝑓𝑓(−3) Y. (x + 5)(x + 1)

Secret Message:

What’s New

SAVE FOR A CAUSE

Thru inspiration instilled by their parents and realization brought by Covid-19 pandemic experience, Neah and Neoh, both Senior High School students decided to save money for a charity cause. Neah has a piggy bank with ₱10.00 initial coins inside. She then decided to save ₱5.00 daily out of her allowance. Meanwhile, Neoh who also has a piggy bank with ₱5.00 initial coin inside decided to save ₱3.00 daily. Given the above situation, answer the following questions:

a. How much money will be saved by Neah and Neah after 30 days? after 365

days or 1 year? their combined savings for one year? b. Is the combined savings enough for a charity donation? Why? c. What values were manifested by the two senior high school students? d. Will you do the same thing these students did? What are the other ways that you can help less fortunate people? e. Do you agree with the statement of Pope John Paul II said that “Nobody is so poor he has nothing to give, and nobody is so rich he has nothing to receive"? Justify your answer. f. What functions can represent the amount of their savings in terms of number of days?

5 CO_Q1_General Mathematic SHS Module 3

What is It

In the previous modules, you learned to represent real life situations to functions and evaluate a function at a certain value. The scenario presented above is an example of real world problems involving functions. This involves two functions representing the savings of the two senior high school students.

Below is the representation of two functions represented by a piggy bank:

Neah Neoh Combined

𝑓𝑓(𝑥𝑥) = 5𝑥𝑥 + 10 𝑔𝑔(𝑥𝑥) = 3𝑥𝑥 + 5 ℎ(𝑥𝑥) = 8𝑥𝑥 + 15

+ =

Suppose that we combine the piggy banks of the two students, the resulting is another piggy bank. It’s just like adding two functions will result to another function.

Definition. Let f and g be functions.

1. Their sum, denoted by 𝑓𝑓 + 𝑔𝑔, is the function denoted by (𝑓𝑓 + 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥).

2. Their difference, denoted by 𝑓𝑓 − 𝑔𝑔, is the function denoted by (𝑓𝑓 − 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) − 𝑔𝑔(𝑥𝑥).

3. Their product, denoted by 𝑓𝑓 • 𝑔𝑔, is the function denoted by (𝑓𝑓 • 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) • 𝑔𝑔(𝑥𝑥).

4. Their quotient, denoted by 𝑓𝑓/𝑔𝑔, is the function denoted by (𝑓𝑓/𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥)/𝑔𝑔(𝑥𝑥), excluding the values of x where 𝑔𝑔(𝑥𝑥) = 0.

5. The composite function denoted by (𝑓𝑓 ° 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑔𝑔(𝑥𝑥)). The process of obtaining a composite function is called function composition.

Example 1. Given the functions:

𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 5 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 − 1 ℎ(𝑥𝑥) = 2𝑥𝑥2 + 9𝑥𝑥 − 5 Determine the following functions: a. (𝑓𝑓 + 𝑔𝑔)(𝑥𝑥) b. (𝑓𝑓 − 𝑔𝑔)(𝑥𝑥) c. (𝑓𝑓 • 𝑔𝑔)(𝑥𝑥) d. (ℎ

𝑔𝑔)(𝑥𝑥)

𝑒𝑒. (𝑓𝑓 + 𝑔𝑔)(3) 𝑓𝑓. (𝑓𝑓 − 𝑔𝑔)(3) 𝑔𝑔. (𝑓𝑓 • 𝑔𝑔)(3)

ℎ. (ℎ𝑔𝑔

)(3)

6 CO_Q1_General Mathematic SHS Module 3

Solution:

𝑎𝑎. (𝑓𝑓 + 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥) definition of addition of functions = (𝑥𝑥 + 5) + (2𝑥𝑥 − 1) replace f(x) and g(x) by the given values = 3𝑥𝑥 + 4 combine like terms b. (𝑓𝑓 − 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) − 𝑔𝑔(𝑥𝑥) definition of subtraction of functions

= (𝑥𝑥 + 5) − (2𝑥𝑥 − 1) replace f(x) and g(x) by the given values = 𝑥𝑥 + 5 − 2𝑥𝑥 + 1 distribute the negative sign = −𝑥𝑥 + 6 combine like terms

c. (𝑓𝑓 • 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) • 𝑔𝑔(𝑥𝑥) definition of multiplication of functions = (𝑥𝑥 + 5) • (2𝑥𝑥 − 1) replace f(x) and g(x) by the given values = 2𝑥𝑥2 + 9𝑥𝑥 − 5 multiply the binomials

d. �ℎ𝑔𝑔� (𝑥𝑥) = ℎ(𝑥𝑥)

𝑔𝑔(𝑥𝑥) definition of division of functions

= 2𝑥𝑥2+9𝑥𝑥−52𝑥𝑥−1

replace h(x) and g(x) by the given values

= (𝑥𝑥+5)(2𝑥𝑥−1)2𝑥𝑥−1

factor the numerator

= (𝑥𝑥+5)(2𝑥𝑥−1)2𝑥𝑥−1

cancel out common factors

= 𝑥𝑥 + 5

e. (𝑓𝑓 + 𝑔𝑔)(3) = 𝑓𝑓(3) + 𝑔𝑔(3) Solve for 𝑓𝑓(3) and 𝑔𝑔(3) separately:

𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 5 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 − 1 𝑓𝑓(3) = 3 + 5 𝑔𝑔(3) = 2(3) − 1 = 8 = 5

∴ 𝑓𝑓(3) + 𝑔𝑔(3) = 8 + 5 = 13

Alternative solution: We know that (𝑓𝑓 + 𝑔𝑔)(3) means evaluating the function (𝑓𝑓 + 𝑔𝑔) at 3.

(𝑓𝑓 + 𝑔𝑔)(𝑥𝑥) = 3𝑥𝑥 + 4 resulted function from item a (𝑓𝑓 + 𝑔𝑔)(3) = 3(3) + 4 replace x by 3 = 9 + 4 multiply = 13 add For item 𝑓𝑓 𝑡𝑡𝑡𝑡 ℎ we will use the values of 𝑓𝑓(3) = 8 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔(3) = 5

f. (𝑓𝑓 − 𝑔𝑔)(3) = 𝑓𝑓(3) − 𝑔𝑔(3) definition of subtraction of functions

= 8 − 5 replace f(3) and g(3) by the given values = 3 subtract

7 CO_Q1_General Mathematic SHS Module 3

Alternative solution:

(𝑓𝑓 − 𝑔𝑔)(𝑥𝑥) = −𝑥𝑥 + 6 resulted function from item b (𝑓𝑓 − 𝑔𝑔)(3) = −3 + 6 replace x by 3 = 3 simplify

g. (𝑓𝑓 • 𝑔𝑔)(3) = 𝑓𝑓(3) • 𝑔𝑔(3) definition of multiplication of functions = 8 • 5 replace f(3) and g(3) by the given values = 40 multiply Alternative solution:

(𝑓𝑓 • 𝑔𝑔)(𝑥𝑥) = 2𝑥𝑥2 + 9𝑥𝑥 − 5 resulted function from item c (𝑓𝑓 • 𝑔𝑔)(3) = 2(3)2 + 9(3) − 5 replace x by 3 = 2(9) + 27 − 5 square and multiply = 18 + 27 − 5 multiply = 40 simplify

h. �ℎ𝑔𝑔� (3) = ℎ(3)

𝑔𝑔(3)

Solve for ℎ(3) and 𝑔𝑔(3) separately: ℎ(𝑥𝑥) = 2𝑥𝑥2 + 9𝑥𝑥 − 5 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 − 1 ℎ(3) = 2(3)2 + 9(3) − 5 𝑔𝑔(3) = 2(3) − 1 = 18 + 27 − 5 = 5 = 40

∴ �ℎ𝑔𝑔� (3) =

ℎ(3)𝑔𝑔(3)

=405

= 8

Alternative solution:

�ℎ𝑔𝑔� (𝑥𝑥) = 𝑥𝑥 + 5 resulted function from item d

�hg� (x) = 3 + 5 replace x by 3

= 8 simplify

Can you follow with what has been discussed from the above examples? Notice that addition, subtraction, multiplication, and division can be both performs on real numbers and functions.

The illustrations below might help you to better understand the concepts on function operations.

In the illustrations, the numbers above are the inputs which are all 3 while below the function machine are the outputs. The first two functions are the functions to be added, subtracted, multiplied and divided while the rightmost function is the resulting function.

8 CO_Q1_General Mathematic SHS Module 3

Addition

Subtraction

Multiplication

Division

Notes to the Teacher

Give emphasis to the students that performing operations on two or more functions results to a new function. The function (𝑓𝑓 + 𝑔𝑔)(𝑥𝑥) is a new function resulted from adding 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥). The new function can now be used to evaluate (𝑓𝑓 + 𝑔𝑔)(3) and it will be the same as adding 𝑓𝑓(3) and 𝑔𝑔(3).

9 CO_Q1_General Mathematic SHS Module 3

Composition of functions:

In composition of functions, we will have a lot of substitutions. You learned in previous lesson that to evaluate a function, you will just substitute a certain number in all of the variables in the given function. Similarly, if a function is substituted to all variables in another function, you are performing a composition of functions to create another function. Some authors call this operation as “function of functions”.

Example 2. Given 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 5𝑥𝑥 + 6, and ℎ(𝑥𝑥) = 𝑥𝑥 + 2

Find the following:

a. (𝑓𝑓 ∘ ℎ)(𝑥𝑥) b. (𝑓𝑓 ∘ ℎ)(4) c. (ℎ ∘ 𝑓𝑓)(𝑥𝑥)

Solution.

a. (𝑓𝑓 ∘ ℎ)(𝑥𝑥) = 𝑓𝑓(ℎ(𝑥𝑥)) definition of function composition

= 𝑓𝑓(𝑥𝑥 + 2) replace h(x) by x+2

Since 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 5𝑥𝑥 + 6 given

𝑓𝑓(𝑥𝑥 + 2) = (𝑥𝑥 + 2)2 + 5(𝑥𝑥 + 2) + 6 replace x by x+2

= 𝑥𝑥2 + 4𝑥𝑥 + 4 + 5𝑥𝑥 + 10 + 6 perform the operations

= 𝑥𝑥2 + 9𝑥𝑥 + 20 combine similar terms

Composition of function is putting a function inside another function. See below figure for illustration.

10 CO_Q1_General Mathematic SHS Module 3

a. (𝑓𝑓 ∘ ℎ)(4) = 𝑓𝑓(ℎ(4)) Step 1. Evaluate ℎ(4) Step 2. Evaluate 𝑓𝑓(6)

ℎ(𝑥𝑥) = 𝑥𝑥 + 2 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 5𝑥𝑥 + 6 ℎ(4) = 4 + 2 𝑓𝑓(6) = 62 + 5(6) + 6 = 6 = 36 + 30 + 6

= 72

(𝑓𝑓 ∘ ℎ)(4) = 𝑓𝑓(ℎ�(4)� = 𝑓𝑓(6) ∴ = 72 To evaluate composition of function, always start with the inside function

(from right to left). In this case, we first evaluated ℎ(4) and then substituted the resulted value to 𝑓𝑓(𝑥𝑥).

Alternative solution:

(𝑓𝑓 ∘ ℎ)(𝑥𝑥)) = 𝑓𝑓(ℎ(𝑥𝑥)) definition of function composition 𝑓𝑓(ℎ(𝑥𝑥)) = 𝑥𝑥2 + 9𝑥𝑥 + 20, from item a (𝑓𝑓 ∘ ℎ)(4)) = 42 + 9(4) + 20 replace all x’s by 4 = 16 + 36 + 20 perform the indicated operations = 72 simplify

A mapping diagram can also help you to visualize the concept of evaluating a function composition.

11 CO_Q1_General Mathematic SHS Module 3

From the definition of function composition, (𝑓𝑓 ∘ ℎ)(4) = 𝑓𝑓(ℎ�(4)�. Looking at the mapping diagram for values and working from right to left, ℎ(4) = 6. Substituting 6 to ℎ(4) we have 𝑓𝑓(6). From the diagram, 𝑓𝑓(6) is equal to 72. Therefore, (𝑓𝑓 ∘ ℎ)(4) =𝑓𝑓(ℎ�(4)� = 72.In the diagram, the first function ℎ(𝑥𝑥) served as the inside function while the second function 𝑓𝑓(𝑥𝑥) is the outside function.

A table of values is another way to represent a function. The mapping diagram above has a corresponding table of values below:

ℎ(𝑥𝑥) = 𝑥𝑥 + 2 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 5𝑥𝑥 + 6

𝑥𝑥 1 3 4 6 ℎ(𝑥𝑥) 3 5 6 8 𝑓𝑓(𝑥𝑥) 12 30 42 72

(𝑓𝑓 ∘ ℎ)(4) = 𝑓𝑓(ℎ�(4)� definition of composition of functions = 𝑓𝑓(6) substitute h(4) by 6 = 72 from the table

b. (ℎ ∘ 𝑓𝑓)(𝑥𝑥) = ℎ(𝑓𝑓(𝑥𝑥)) definition of composition of functions = ℎ(𝑥𝑥2 + 5𝑥𝑥 + 6), substitute f(x) by x2 + 5x + 6, given Since ℎ(𝑥𝑥) = 𝑥𝑥 + 2 given ℎ(𝑥𝑥2 + 5𝑥𝑥 + 6) = 𝑥𝑥2 + 5𝑥𝑥 + 6 + 2 substitute x by x2 + 5x + 6 = 𝑥𝑥2 + 5𝑥𝑥 + 8 combine similar terms

Notes to the Teacher

The functions (𝑓𝑓 ∘ ℎ)(𝑥𝑥) and (h ∘ f)(x) are generally not the same as we see in the previous examples. It only means that order of functions counts in composition of function operation. There are special cases where they will be the same; this is when the two functions are inverses. Graphing and finding the domain and range of algebraic operations is not covered by this module but this is an interesting activity that can be used as enrichment once this module was mastered.

12 CO_Q1_General Mathematic SHS Module 3

What’s More

Activity 1:

MATCHING FUNCTIONS Match column A with column B by writing the letter of the correct answer on the blank before each number

Given: 𝑎𝑎(𝑥𝑥) = 𝑥𝑥 + 2 𝑏𝑏(𝑥𝑥) = 5𝑥𝑥 − 3

𝑐𝑐(𝑥𝑥) =𝑥𝑥 + 5𝑥𝑥 − 7

𝑎𝑎(𝑥𝑥) = √𝑥𝑥 + 5

𝑒𝑒(𝑥𝑥) =3

𝑥𝑥 − 7

Column A Column B ______1. (𝑎𝑎 + 𝑏𝑏)(𝑥𝑥) a. 3

𝑥𝑥+5

______2. (𝑎𝑎 • 𝑏𝑏)(𝑥𝑥) b. ±3 ______3. (𝑎𝑎 ∘ 𝑎𝑎)(𝑥𝑥) c. −7 ______4. �𝑒𝑒

𝑐𝑐� (𝑥𝑥) d.−4

5

______5. (𝑐𝑐 − 𝑒𝑒)(𝑥𝑥) e. √𝑥𝑥 + 7 ______6. (𝑎𝑎 + 𝑏𝑏)(−1) f. 𝑥𝑥+2

𝑥𝑥−7

______7. (𝑎𝑎 • 𝑏𝑏)(0) g. 6𝑥𝑥 − 1 ______8. (𝑎𝑎 ∘ 𝑎𝑎)(2) h. 1 ______9. �𝑒𝑒

𝑐𝑐� (−2) i. −6

______10. (𝑐𝑐 − 𝑒𝑒)(2) j. 5𝑥𝑥2 + 7𝑥𝑥 − 6

Activity 2:

LET’S SIMPLIFY

A. Let 𝑝𝑝(𝑥𝑥) = 2𝑥𝑥2 + 5𝑥𝑥 − 3,𝑚𝑚(𝑥𝑥) = 2𝑥𝑥 − 1,𝑎𝑎𝑎𝑎𝑎𝑎 ℎ(𝑥𝑥) = 𝑥𝑥+1𝑥𝑥−2

Find: 1. (𝑚𝑚− 𝑝𝑝)(𝑥𝑥) 2. 𝑝𝑝(5) + 𝑚𝑚(3) − ℎ(1) 3. 𝑚𝑚(𝑥𝑥)

𝑝𝑝(𝑥𝑥)

4. 𝑝𝑝(𝑥𝑥 + 1) 5. 𝑝𝑝(3) − 3(𝑚𝑚(2)

13 CO_Q1_General Mathematic SHS Module 3

B. Given the following: • 𝑚𝑚(𝑥𝑥) = 5𝑥𝑥 − 3 • 𝑎𝑎(𝑥𝑥) = 𝑥𝑥 + 4 • 𝑐𝑐(𝑥𝑥) = 5𝑥𝑥2 + 17𝑥𝑥 − 12 • 𝑡𝑡(𝑥𝑥) = 𝑥𝑥−5

𝑥𝑥+2

Determine the following functions.

1. (𝑚𝑚 + 𝑎𝑎)(𝑥𝑥) 2. (𝑚𝑚 ∙ 𝑎𝑎)(𝑥𝑥) 3. (𝑎𝑎 − 𝑐𝑐)(𝑥𝑥) 4. (𝑐𝑐/𝑚𝑚)(𝑥𝑥) 5. (𝑚𝑚 ∘ 𝑎𝑎)(𝑥𝑥) 6. (𝑎𝑎 ∘ 𝑐𝑐)(−3) 7. 𝑎𝑎(𝑚𝑚�𝑚𝑚(2)�)

C. Given the functions 𝑔𝑔(𝑥𝑥) = 𝑥𝑥2 − 4 and ℎ(𝑥𝑥) = 𝑥𝑥 + 2, Express the following as the sum, difference, product, or quotient of the functions above. 1. 𝑝𝑝(𝑥𝑥) = 𝑥𝑥 − 2 2. 𝑟𝑟(𝑥𝑥) = 𝑥𝑥2 + 𝑥𝑥 − 2 3. 𝑠𝑠(𝑥𝑥) = 𝑥𝑥3 + 2𝑥𝑥2 − 4𝑥𝑥 − 8 4. 𝑡𝑡(𝑥𝑥) = −𝑥𝑥2 + 𝑥𝑥 + 6

D. Answer the following:

1. Given ℎ(𝑥𝑥) = 3𝑥𝑥2 + 2𝑥𝑥 − 4,𝑤𝑤ℎ𝑎𝑎𝑡𝑡 𝑓𝑓𝑠𝑠 ℎ(𝑥𝑥 − 3)?

2. Given 𝑎𝑎(𝑥𝑥) = 𝑥𝑥 + 5 𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝(𝑥𝑥) = 𝑥𝑥2 + 3𝑥𝑥 − 10,𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎: a. (𝑎𝑎 − 𝑝𝑝)(𝑥𝑥) + 3𝑝𝑝(𝑥𝑥)

b. 𝑛𝑛(𝑥𝑥)𝑝𝑝(𝑥𝑥)

c. (𝑝𝑝 ∘ 𝑎𝑎)(𝑥𝑥) 3. Let 𝑚𝑚(𝑥𝑥) = √𝑥𝑥 + 3, 𝑎𝑎(𝑥𝑥) = 𝑥𝑥3 − 4, 𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝(𝑥𝑥) = 9𝑥𝑥 − 5,𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎 �𝑚𝑚 ∘ (𝑎𝑎 − 𝑝𝑝)�(3). 4. Given 𝑤𝑤(𝑥𝑥) = 3𝑥𝑥 − 2, 𝑣𝑣(𝑥𝑥) = 2𝑥𝑥 + 7 and 𝑘𝑘(𝑥𝑥) = −6𝑥𝑥 − 7, find (𝑤𝑤 − 𝑣𝑣 − 𝑘𝑘)(2) 5. If 𝑠𝑠(𝑥𝑥) = 3𝑥𝑥 − 2 and 𝑟𝑟(𝑥𝑥) = 2

𝑥𝑥+5, find 2(𝑠𝑠 + 𝑟𝑟)(𝑥𝑥)

6. Given 𝑎𝑎(𝑥𝑥) = 4𝑥𝑥 + 2, 𝑏𝑏(𝑥𝑥) = 32𝑥𝑥,𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐(𝑥𝑥) = 𝑥𝑥 − 5,𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎 (𝑎𝑎 • 𝑏𝑏 • 𝑐𝑐)(𝑥𝑥)

14 CO_Q1_General Mathematic SHS Module 3

What I Have Learned

Complete the worksheet below with what have you learned regarding operations on functions. Write your own definition and steps on performing each functions operation. You may give your own example to better illustrate your point.

What I Can Do

Read and understand the situation below, then answer the questions that follow.

The outbreak of coronavirus disease 2019 (COVID-19) has created a global health crisis that has had a deep impact on the way we perceive our world and our everyday lives, (https://www.frontiersin.org).Philippines, one of the high-risk countries of this pandemic has recorded high cases of the disease. As a student, you realize that Mathematics can be a tool to better assess the situation and formulate strategic plan to control the disease.

Suppose that in a certain part of the country, the following data have been recorded.

𝑎𝑎 0 1 2 3 4 5 6 7 8

𝐼𝐼(𝑎𝑎) 3 5 9 12 18 25 35 47 82

Where I(d) is the function of the number of people who got infected in d days

Addition Subtraction Multiplication Division Composition

15 CO_Q1_General Mathematic SHS Module 3

The number of recoveries was also recorded in the following table as the function 𝑅𝑅(𝑓𝑓) where R as the number of recoveries is dependent to number of infected (I).

𝐼𝐼 3 5 9 12 18 25 35 47 82

𝑅𝑅(𝐼𝐼) 0 1 2 5 7 9 12 18 25

a. Evaluate the following and then interpret your answer. 1. 𝑅𝑅(𝐼𝐼(3)) 2. 𝑅𝑅(𝐼𝐼(8)) 3. 𝐼𝐼(𝑅𝑅(18))

b. The number of deaths (M) was also dependent on the number of infected

(I). Complete the table with your own number of deaths values for the given number of infected.

𝐼𝐼 3 5 9 12 18 25 35 47 82

𝑀𝑀(𝐼𝐼) 0 0 1 1 1 2 3 4 6

Evaluate the following and then interpret your answer.

1. 𝑀𝑀(𝐼𝐼(1)) 2.𝑀𝑀(𝐼𝐼(4)) 3. 𝐼𝐼(𝑀𝑀(12))

c. What can you conclude about the data presented?

d. What can you suggest to the government to solve the problem?

16 CO_Q1_General Mathematic SHS Module 3

Assessment

Write the letter of the correct answer on a separate answer sheet.

1. The following are notations for composite functions EXCEPT, a. ℎ(𝑝𝑝(𝑥𝑥)) c. (𝑠𝑠 ∘ 𝑡𝑡)(𝑥𝑥) b. b. 𝑓𝑓(𝑥𝑥)𝑔𝑔(𝑥𝑥) d. 𝑓𝑓(𝑔𝑔(𝑥𝑥))

2. Find ℎ(3) + 𝑎𝑎(2) 𝑓𝑓𝑓𝑓 ℎ(𝑥𝑥) = 𝑥𝑥 − 1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎(𝑥𝑥) = 7𝑥𝑥 + 3

a. 2 c. 14 b. b. 5 d. 19

3. 𝑡𝑡(𝑥𝑥) = −𝑥𝑥2 + 7𝑥𝑥 + 1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟(𝑥𝑥) = 5𝑥𝑥2 − 2 𝑥𝑥 + 8, 𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎 (𝑡𝑡 − 𝑟𝑟)(2).

a. 18 c. -13 b. b. -18 d. 13

4. 𝑓𝑓(𝑥𝑥) = 4𝑥𝑥 + 2 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔(𝑥𝑥) = 3𝑥𝑥 − 1,𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎 (𝑓𝑓 − 𝑔𝑔)(4).

a. 0 c. 7 b. -9 d. -8

5. 𝐼𝐼𝑓𝑓 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 4 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 5 𝐹𝐹𝑓𝑓𝑎𝑎𝑎𝑎 𝑓𝑓(𝑥𝑥) • 𝑔𝑔(𝑥𝑥) a. 𝑥𝑥2 + 𝑥𝑥 + 20 c. 𝑥𝑥2 – 𝑥𝑥 − 20 b. 𝑥𝑥2 – 𝑥𝑥 + 20 d. 𝑥𝑥2 + 𝑥𝑥 − 20

6. Given ℎ(𝑎𝑎) = 𝑛𝑛+6𝑛𝑛−4

𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝(𝑘𝑘) = 𝑛𝑛+6𝑛𝑛2+4𝑛𝑛−32

. Find ℎ𝑝𝑝

(𝑘𝑘).

a. 1𝑛𝑛+8

c. 1𝑛𝑛−8

b. b. 𝑎𝑎 − 8 d. 𝑎𝑎 + 8

7. If 𝑓𝑓(𝑥𝑥) = 18𝑥𝑥2 and 𝑡𝑡(𝑥𝑥) = 8𝑥𝑥, find 𝑓𝑓

𝑡𝑡(𝑥𝑥).

a. 9𝑥𝑥4 c. 4

9𝑥𝑥

b. b. 4𝑥𝑥9 d. 9

4𝑥𝑥

8. When 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 − 5 and 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥2 − 5 , find 𝑓𝑓(𝑔𝑔(𝑥𝑥)).

a. 𝑥𝑥2 + 2𝑥𝑥 + 3 c. 6𝑥𝑥2 + 20 b. b. 6𝑥𝑥2 − 20 d. 2𝑥𝑥2 + 6

9. 𝑟𝑟(𝑥𝑥) = 𝑥𝑥 + 5 and 𝑞𝑞(𝑥𝑥) = 2𝑥𝑥2 − 5, Find 𝑞𝑞(𝑟𝑟(−2))

a. 8 c. 13 b. b. -8 d. -13

10. Let 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 + 8 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2. Find 𝑓𝑓(𝑔𝑔(𝑥𝑥)).

                a.  2𝑥𝑥 + 3 c. 4𝑥𝑥 + 1 b. 2𝑥𝑥 − 3 d. 3𝑥𝑥 + 2

17 CO_Q1_General Mathematic SHS Module 3

For numbers 11-13, refer to the figure below:

11. Evaluate 𝑟𝑟(2) a. -11 c. 5 b. -3 d. 11

12. Find 𝑠𝑠(𝑟𝑟(7)) a. 7 c. -1 b. 1 d. -7

13. Find (𝑠𝑠 ∘ 𝑟𝑟)(1) a. -3 c. 5 b. 3 d. -5

For numbers 14-15, refer to the table of values below

𝑡𝑡(𝑥𝑥) = 2𝑥𝑥 + 1 𝑘𝑘(𝑥𝑥) = 2𝑥𝑥2 − 7𝑥𝑥 − 5

𝒙𝒙 0 1 2 3 4 5 6 7 8

𝑡𝑡(𝑥𝑥) 1 3 5 7 9 11 13 15 17

𝑘𝑘(𝑥𝑥) -5 -10 -11 -8 -1 10 25 44 67

14. Find (𝑘𝑘 − 𝑡𝑡)(4)

a. 8 c. 10 b. -8 d. -10

15. Find (𝑘𝑘 ∘ 𝑡𝑡)(2) a. 10 c. -5 b. b. -10 d. -1

18 CO_Q1_General Mathematic SHS Module 3

Additional Activities

PUNCH D LINE

Find out some of favorite punch lines by answering operations on functions problems below. Phrases of punch lines were coded by the letters of the correct answers. Write the punch lines on the lines provided.

Given:

𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 − 1𝑔𝑔(𝑥𝑥) = |3𝑥𝑥 − 4|ℎ(𝑥𝑥) =𝑥𝑥2

𝑟𝑟(𝑥𝑥) = 𝑥𝑥 + 3𝑠𝑠(𝑥𝑥) = 𝑥𝑥2 − 4𝑥𝑥 − 21

Column A Column B

_______1.𝑓𝑓(0) = A. −11

_______2. 𝑔𝑔(3) = B. 2

_______3. 𝑠𝑠(−1) = C. 3𝑥𝑥 + 2

_______4. ℎ(0) = D. 𝑥𝑥 − 7

_______5. (𝑓𝑓 + 𝑟𝑟)(𝑥𝑥) = E. −𝑥𝑥 + 4

_______6. (𝑓𝑓 + 𝑟𝑟)(3) = F. 0

_______7. (𝑟𝑟 − 𝑓𝑓)(𝑥𝑥) = G. 2𝑥𝑥2 + 5𝑥𝑥 − 3

_______8.(𝑟𝑟 − 𝑓𝑓)(2) = H. 6

_______9. (𝑓𝑓 • 𝑟𝑟)(𝑥𝑥) = I. −16

_______10. (𝑓𝑓 • 𝑟𝑟)(1) = J. 2𝑥𝑥 + 2

_______11. 𝑠𝑠𝑟𝑟

(𝑥𝑥) = K. 5

_______12. 𝑠𝑠𝑟𝑟

(−4) = L. 1

_______13. (𝑟𝑟 ○ 𝑓𝑓)(𝑥𝑥) = M. 11

_______14. (𝑟𝑟 ○ 𝑓𝑓)(2) = N. −1

_______15. (𝑔𝑔 ○ 𝑓𝑓)(1) = O. 4

19 CO_Q1_General Mathematic SHS Module 3

Code:

Tingnan mo ako K ang laman ng utak ko? J

para may attachment lagi tayo L Buhay nga pero patay I

ang parents ko E Hindi lahat ng buhay ay buhay N

na ako sa’yo O Di mo pa nga ako binabato B

Masasabi mo bang bobo ako? D na patay naman sa’yo F

Kasi, botong-boto sayo M Tatakbo ka ba sa eleksyon? C

Kung ikaw lamang A pero tinamaan G

Sana naging email na lang ako H

Punch lines:

(1-4) ___________________________________________________________ (5-7) ___________________________________________________________ (8-10) ___________________________________________________________ (11-13) ___________________________________________________________ (14-15) ___________________________________________________________

20 CO_Q1_General Mathematic SHS Module 3

Answer Key

21 CO_Q1_General Mathematic SHS Module 3

22 CO_Q1_General Mathematic SHS Module 3

References Department of Education. "General Mathematics Learner's Material." In General

Mathematics Learner's Material, by Debbie Marie B. Verzosa, Paolo L. Apolinario, Regina M. Tresvalles, Francis Nelson M. Infante, Jose Lorenzo M. Sin and Len Patrick Dominic M. Garces, edited by Leo Andrei A. Crisologo, Shirlee R. Ocampo, Jude Buot, Lester C. Hao, Eden Delight P. Miro and Eleanor Villanueva, 13-20. Meralco Avenue, Pasig City, Philippines 1600: Lexicon Pres Inc., 2016.

Department of education. "General Mathematics Teacher's Guide." In General Mathematics Teacher's Guide, by Leo Andrei A. Crisologo, Shirlee R. Ocampo, Eden Delight P. Miro, Regina M. Tresvalles, Lester C. Hao and Emellie G. Palomo, edited by Christian Paul O. Chan Shio and Mark L. Loyola, 14-22. Meralco Avenue, Pasig City, Philippines 1600: Lexicon Press Inc., 2016.

coronatracker.com. COVID-19 Corona Tracker. n.d. https://www.coronatracker.com/country/philippines/ (accessed May 20, 2020).

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