The document provides information about functions and relations, including definitions and examples of representing real-world situations with functions. It discusses writing function rules from tables of data and representing relationships between variables with equations. Examples are given of determining whether relations qualify as functions based on their ordered pairs, tables of values, or mapping diagrams.

1. i
General
Mathematics
Quarter 1 – Module 1
Functions
Department of Education Republic of the Philippines
SENIOR HIGH SCHOOL

2. ii
11
General
Mathematics
Module 1: INTRODUCTION TO
FUNCTIONS
Department of Education • Republic of the Philippines
This instructional material was collaboratively developed and reviewed
by educators from public and private schools, colleges, and/or universities.
We encourage teachers and other education stakeholders to email their
feedback, comments, and recommendations to the Department of Education
at action@deped.gov.ph.
We value your feedback and recommendations.

3. iii
DEVELOPMENT TEAM OF THE MODULE
Authors: Edward C. Reyes Jr.
Editors:
Illustrator:
Layout Artist:
Management Team
Chairperson: Dr. Arturo B. Bayocot, CESO III
Regional Director
Co-Chairpersons: Dr. Victor G. De Gracia Jr. CESO V
Assistant Regional Director
Jonathan S. dela Peña, PhD, CESO V
Schools Division Superintendent
Rowena H. Para-on, PhD
Assistant Schools Division Superintendent
Mala Epra B. Magnaong, Chief ES, CLMD
Members: Neil A. Improgo, PhD, EPS-LRMS; Bienvenido U. Tagolimot, Jr., PhD, EPS-
ADM; Erlinda G. Dael, PhD, CID Chief; Nelson B. Absin, PhD, EPS (Math &
Science); Celieto B. Magsayo, LRMS Manager; Loucile L. Paclar, Librarian II;
Kim Eric G. Lubguban, PDO II
Regional Evaluator: Maria Jocelyn Y. Aguiman
Camiguin Division
General Mathematics – Grade 11
Alternative Delivery Mode
Module 1: Introduction to Functions
First Edition, 2019
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, poems, pictures, photos, brand names,
trademarks, etc.) included in this book 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– Region X – Northern Mindanao.
Printed in the Philippines by ______________________________________
Department of Education – Bureau of Learning Resources (DepEd-BLR)
Office Address:
Telefax:
E-mail Address:

4. iv
TABLE OF CONTENTS
Overview …………………………………………………………………………………1
Module Content ……………………………………………………………………………..1
Objectives ………………………………………………………………………………...1
General Instructions…………………………………………………………………………2
Pretest………………………………………………………………………………………...3
Lesson 1: Representations of Functions and Relations ……………………………4
Activity 1…………………………………………………………………… 14
Lesson 2: Evaluating Function .………………………………………………………….16
Activity 2……………………………………………………………………..18
Lesson 3: Operations on Function ………………………………………………………20
Composition of Functions …………………………………………………... 24
Problems involving Functions ……………………………………………….25
Activity 3…………………………………………………………………….25
Summary/Generalizations………………………………………………………………...27
Posttest……………………………………………………………………………………...28
References………………………………………………………………………………….

5. 1
What I need to Know
Module Content
In this module, you will learn to:
1. represent real-life situations using functions, including piece-wise functions;
2. evaluate a function;
3. performs addition, subtraction, multiplication, division and composition of
functions;
and
4. solves problems involving functions.
Dear learner,
Welcome to Module 1 for General Mathematics!
In this module, the competencies expected that you will learn are found in the
Module Content. You will see how relations and functions are represented and what
piece-wise functions are. You will also learn how to evaluate perform operations with
functions and composite functions. Plus, you will need critical thinking skills as you
solve problems with functions.
However, can you do the PRE-TEST?
You may then start this module. Try to understand the Lesson 1 and Lesson 2,
learn from the illustrative and solved examples, and do the activities (Activity 1 to
Activity 6). Take the challenge in the Posttest. Then, check your work. Answers
are provided in the ANSWER KEY. Read the Summary and generalizations.
For sure, you will enjoy learning how to represent relations and functions. Do
not hesitate to ask help from your teacher if there are difficulties that you have
encountered.
Good Luck!

6. 2
General Directions
To help you attain the objectives of this module, you may try following the
steps below.
 First, read carefully each lesson on this module. Should there be times that
you need to read again parts of the lesson, go ahead!
 Second, answer the pre-assessment test. It is expected that some parts may
be unfamiliar to you as new lessons will be learned in this module.
 Third, read and follow instructions honestly.
 Fourth, do not hesitate to answer all the activities set for you. Your teacher
will be glad to answer your queries.
 Then, you may check answers to each activity. An Answer Key is provided.
 And lastly, read the Summary carefully so you will not miss out important
concepts in this module.
What I Know
Let us check how much you know about functions and their graphs.
Direction: Choose the letter of the best answer and write this on your answer sheet.
1) Given 𝑓(𝑥) = 2𝑥 − 5 & 𝑔(𝑥) = 3𝑥 + 4, solve for (𝑔 ○ 𝑓)(𝑥).
a. 11 − 6𝑥 c. 6𝑥 − 11
b. 6𝑥2
− 7𝑥 − 20 d. 6𝑥2
− 23𝑥 − 20
2) Given 𝑦 = 3𝑥 + 7, what is 𝑓(−2)?
a. 1 c. -13
b. -1 d. 13
3) The composite function denoted by 𝑓 ○ 𝑔 is defined as _____________.
a. (𝑓 ○ 𝑔)(𝑥) = 𝑓(𝑔(𝑥)) c. (𝑓 ○ 𝑔)(𝑥) = 𝑓(𝑥)●𝑔(𝑥)
b. (𝑓 ○ 𝑔)(𝑥) = 𝑔(𝑓(𝑥)) d. (𝑓 ○ 𝑔)(𝑥) = 𝑔(𝑥)●𝑓(𝑥)
4) It is a set of ordered pairs (𝑥, 𝑦) such that no two ordered pairs have the same x-
value but different y-values.
a. relation c. domain
b. function d. range
5) What is the domain of the equation 𝑦 = 3𝑥2
− 4𝑥?
a. {𝒙: 𝒙 ∈ 𝑹, 𝒙 < −𝟏} c. {𝒙: 𝒙 ∈ 𝑹}
b. {𝒙: 𝒙 ∈ 𝑹, 𝒙 ≠ 𝟏} d. {𝒙: 𝒙 ∈ 𝑹, 𝒙 ≥ 𝟒}

7. 3
Answer key on page 31
6) Given 𝑓(𝑥) = 2𝑥 − 5 & 𝑔(𝑥) = 3𝑥 + 4, find (𝑓●𝑔)(𝑥).
a. 6𝑥2
+ 23𝑥 − 20 c. 6𝑥2
− 20
b. 6𝑥2
− 23𝑥 − 20 d. 6𝑥2
− 7𝑥 − 20
7) If 𝑓(𝑥) = 𝑥 + 7 & 𝑔(𝑥) = 2𝑥 − 3, what is (𝑓 − 𝑔)(𝑥)?
a. −𝑥 + 4 c. 𝑥 − 4
b. 10 − 𝑥 d. 10 + 3𝑥
8) When dividing two fractions or rational expressions, multiply the dividend with
the ________ of the divisor.
a. reciprocal c. abscissa
b. addend d. Theorem
9) What is the set of all possible values that the variable x can take in a relation?
a. domain c. equation
b. range d. function
10) Which of the following set of ordered pairs in NOT a function?
a. (1,2), (2,3), (3,4), (4,5) c. (1, 1), (2, 2), (3, 3), (4, 4)
b. (1,2), (1,3), (3,6), (4,8 d. (3, 2), (4, 2), (5, 2), (6, 2)

8. 4
LESSON
1 REPRESENTATIONS OF FUNCTIONS AND RELATIONS
Here you’ll learn how to interpret situations that occur in everyday life and use
functions to represent them. You’ll also use these functions to answer questions that
come up.
What if your bank charged a monthly fee of $15 for your checking account
and also charged $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? In this Concept, you’ll learn how to handle situations like these by using
functions.
How can challenging problems involving functions be analyzed and solved?
Let’s answer these question by doing the activities below.
Activity 1: Pictures Analysis (eliciting prior knowledge, Motivation, Hook)
Observe the pictures below and answer the questions
1. What concepts of functions can you associate with the pictures?
____________________________________________________
2. How these concepts are used indifferent situations?

9. 5
____________________________________________________
3. Can you determine any purpose why these concepts are present in the
pictures? Please specify.
____________________________________________________
4. Can you cite any problem which can be answered through these concepts?
Describe at least one.
____________________________________________________
5. How can challenging problems involving functions be analyzed and solved?
____________________________________________________
Activity 2: IRF- Initial, Revised, Final
How can challenging problems involving functions be analyzed and solved?
Initial Answer Revised Answer Final Answer
Write a Function Rule
In many situations, data is collected by conducting a survey or an experiment. To
visualize the data, it is arranged into a table. Most often, a function rule is needed to
predict additional values of the independent variable.
Example
Try to notice the trend of each variable.
Number of CDs 2 4 6 8 10
Cost (Php) 24 48 72 96 120
Solution:

10. 6
You pay Php 24 for 2 CDs, Php 48 for 4 CDs, and Php 120 for 10 CDs. That
means that each CD costs Php 12.
We can write the function rule.
𝐶𝑜𝑠𝑡 = 𝑃ℎ𝑝 12 × 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝐷𝑠 or 𝒇(𝒙) = 𝟏𝟐𝒙
Example
Write a
function rule for the table.
Solution:
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: 𝒇(𝒙) = |𝒙|.
Represent a Real-World Situation with a Function.
Let’s look at a real-world situation that can be represented by a function.
Example
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.
Solution:
Let 𝑥 = the number of hours Maya spends on the internet in one month.
𝑦 = Maya’s monthly cost.
The monthly fee is $11.95 with an hourly charge of $0.50.
The total cost = flat fee + hourly fee × number of hours. The function is
𝒚 = 𝒇(𝒙) = 𝟏𝟏. 𝟗𝟓 + 𝟎. 𝟓𝟎𝒙.
𝒙 −𝟐 𝟎 𝟐 −𝟑 −𝟏 𝟏 𝟑
𝒚 𝟐 𝟎 𝟐 𝟑 𝟏 𝟏 𝟑

11. 7
Definition
A relation is a rule that relates values from a set of values (called the domain) to a second set
of values (called the range).
A relation is a set of ordered pairs (𝑥, 𝑦).
A function is a relation where each element in the domain is related to only one value in the
range by some rule.
A function is a set of ordered pairs (𝑥, 𝑦) such that no two ordered pairs have the same x-value
but different y-values. Using functional notation, we can write 𝑓(𝑥) = 𝑦, read as
“𝑓 𝑜𝑓 𝑥 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑦. ” In particular, if (1, 2) is an ordered pair associated with the function f,
then we say that 𝑓(2) = 1.
Here is a video to introduce functions https://www.youtube.com/watch?v=tAoe4xjUZQk
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.
a. Write a function expressing how pressure changes depending on depth
underwater.
b. How far can you dive without experiencing more than 58.7 psi of pressure on your
body?
Process Questions:
1. How did you answer the problem above?
2. What concept did you use to solve the problem?
3. What might happen if you can’t be able to respond to the given situation?
4. How can challenging problems involving geometric figures be analyzed and
solved?
Write your answers here:
.

12. 8
What’s More
Relations can be represented by using ordered pairs, graph, table of values,
mapping diagram and rule or equations. Determine which of the following represents
functions.
1. Ordered Pairs
Example 1. Which of the following relations are functions?
𝑓 = (1, 3), (4, 1), (2, 0), (7,2)
𝑔 = (3, 2), (4,4), (3, 3), (8, 9)
ℎ = (1, 2), (2, 3), (3, 4), (4, 5)
Solution:
The relations 𝑓 and ℎ are functions because no two ordered pairs have the
same x-value but different y-values. Meanwhile, 𝑔 is not a function because
(3,2) and (3, 3) are ordered pairs with the same x-value but different y-
values.
Relations and functions can be represented by mapping diagrams where
the elements of the domain are mapped to the elements of the range using
arrows. In this case, the relation or function is represented by the set of all
the connections represented by the arrows.
2. Table of values
Example 2
Answer: Function. This is a many-to- one correspondence.
x -3 -2 -1 0 1 3 4
y 10 5 2 1 2 5 6
x 1 1 1 2 4
A.

13. 9
The Vertical Line Test
A graph represents a function if and only if each vertical line intersects the graph
at most once.
Inspecting the abscissas in the
table,
Answer: mere relation. This is a one- to- many correspondence. Looking at
the table, there is duplication in the domain. The element “1” in x is matched to
three elements in y.
3. Mapping Diagrams
Example 3. Which of the following mapping diagrams represent
functions?
Solution.
The relations f and g are functions because each value y in Y is unique for
a specific value of x. The relation h is not a function because there is at
least one element in X for which there is more than one corresponding y-
value. For example, 𝑥 = 2 corresponds to 𝑦 = 20 or 40.
A relation between two sets of numbers can be illustrated by a graph in the
Cartesian plane, and that a function passes the vertical line test.
Example 4. Which of the following can be graphs of functions?
y 1 2 3 4 5
𝑓 𝑔
ℎ
B.

14. 10
1. 2.
3. 4.
5.
Solution.
Graphs 2, 3, 4 are graphs of functions while 1 and 5 are not because they
Important Concepts.
Relations are rules that relate two values, one from a set of inputs and the second from the set
of outputs.
Functions are rules that relate only one value from the set of outputs to a value from the set
of inputs.
The domain of a relation is the set of all possible values that the variable x can take.

15. 11
do not pass the vertical line test.
Example 5.
Identify the domain for each relation using set builder notation.
a. 𝑦 = 3𝑥 − 2
b. 𝑦 = 3𝑥2
− 4𝑥
c. 𝑥2
+ 𝑦2
= 1
d. 𝑦 = √𝑥 − 4
e. 𝑦 =
2𝑥+1
𝑥−1
f. 𝑦 = ⌊𝑥⌋ + 1 where is the greatest integer function.
Solution. The domains for the relations are as follows:
a. {𝒙: 𝒙 ∈ 𝑹} d. {𝒙: 𝒙 ∈ 𝑹, 𝒙 ≥ 𝟒}
b. {𝒙: 𝒙 ∈ 𝑹} e. {𝒙: 𝒙 ∈ 𝑹, 𝒙 ≠ 𝟏}
c. {𝒙: 𝒙 ∈ 𝑹, −𝟏 ≤ 𝒙 ≤ 𝟏} f. {𝒙: 𝒙 ∈ 𝑹}
Functions as representations of real-life situations.
Functions can often be used to model real situations. Identifying an appropriate
functional model will lead to a better understanding of various phenomena.
Example 6.
Give a function C that can represent the cost of buying x meals, if one meal
costs P40.
Solution: Since each meal costs P40, then the cost function is 𝐶(𝑥) = 40𝑥.

16. 12
Example 7.
One hundred meters of fencing is available to enclose a rectangular area
next to a river (see figure). Give a function A that can represent the area that
can be enclosed, in terms of x.
Solution.
The area of the rectangular enclosure is 𝐴 = 𝑥𝑦. We will write this as a
function of 𝑥. Since only 100 m of fencing is available, then 𝑥 + 2𝑦 = 100
or 𝑦 =
100−𝑥
2
= 50 – 0.5𝑥. Thus, 𝐴 = 𝑥(50 – 0.5𝑥) = 50𝑥 – 0.5𝑥2
.
Piecewise Functions.
Some situations can only be described by more than one formula, depending on the
value of the independent variable.
Example 8.
A user is charged 𝑃300 monthly for a particular mobile plan, which includes
100 free text messages. Messages in excess of 100 are charged P1 each.
Represent the monthly cost for text messaging using the function 𝑡(𝑚),
where m is the number of messages sent in a month.
Solution. The cost of text messaging can be expressed by the piecewise function
𝑡(𝑚) = {
300 , 𝑖𝑓 0 < 𝑚 ≤ 100
300 + 𝑚 , 𝑖𝑓 𝑚 > 100
Example 9.
A jeepney ride costs P8.00 for the first 4 kilometers, and each additional
integer kilometer adds P1.50 to the fare. Use a piecewise function to
represent the jeepney fare in terms of the distance (d) in kilometers.
Solution.

17. 13
The input value is distance and the output is the cost of the jeepney fare. If
𝐹(𝑑) represents the fare as a function of distance, the function can be
represented as follows:
𝐹(𝑑) = {
8.00 , 𝑖𝑓 0 < 𝑑 ≤ 4
8 + 1⌊𝑑⌋ , 𝑖𝑓 𝑑 > 4
Note that ⌊𝑑⌋ is the floor function applied to d. The floor function gives the
largest integer less than or equal to d, e.g. ⌊4.1⌋ = ⌊4.9⌋ = ⌊4⌋
Example 10.
Water can exist in three states: solid ice, liquid water, and gaseous water
vapor. As ice is heated, its temperature rises until it hits the melting point of
0°C and stays constant until the ice melts. The temperature then rises until it
hits the boiling point of 100°C and stays constant until the water evaporates.
When the water is in a gaseous state, its temperature can rise above 100°C
(This is why steam can cause third degree burns!).
A solid block of ice is at -25°C and heat is added until it completely turns into
water vapor. Sketch the graph of the function representing the temperature of
water as a function of the amount of heat added in Joules given the following
information:
 The ice reaches 0°C after applying 940 J.
 The ice completely melts into liquid water after applying a total of 6,950 J.
 The water starts to boil (100°C) after a total of 14,470 J.
 The water completely evaporates into steam after a total of 55,260 J.
Assume that rising temperature is linear. Explain why this is a piecewise function.
Solution. Let 𝑇(𝑥) represent the temperature of the water in degrees Celsius as a
function of cumulative heat added in Joules. The function T(x) can be graphed as
follows:

18. 14
This is a piecewise function because the temperature rise can be expressed as a
linear function with positive slope until the temperature hits 0°C, then it becomes a
constant function until the total heat reaches 6,950𝐾 𝐽. It then becomes linear again
until the temperature reaches 100°C, and becomes a constant function again until
the total heat reaches 55,260 𝐽.
Are you ready to take the test? Right on the next page…
What’s New
Answer the following item as instructed. Write your answer on a separate sheet.
Justify your answer.
Activity 1: RELATION-ships
1. For which values of k is the set of order pairs (2, 4), (𝑘, 6), (4, 0 ) a function?
2. Which of the following diagram represents a relation that is NOT a function?
Congratulations! You have finished the whole lesson.

19. 15
3. Give the domain of 𝑦 = √6 − 𝑥 using set builder notation.
4. A person is earning P600 per day to do a certain job. Express the total salary
S as a function of the number n of days that the person works.
5. A taxi ride costs P40.00 for the first 500 meters, and each additional 300
meters (or a fraction thereof) adds P3.50 to the fare. Use a piecewise function
to represent the taxi fare in terms of the distance d in meters
6. A certain chocolate bar costs P35.00 per piece. However, if you buy more
than 10 pieces, they will be marked down to a price of P32.00 per piece. Use
a piecewise function to represent the cost in terms of the number of chocolate
bars bought.
What I Learned…
1. What did you discover from the activity?
_____________________________________________________________
2. What conjecture or conclusion can you give from what you have learned?
_____________________________________________________________
3. How will you validate your answer?
_____________________________________________________________
4. Be ready to share what you discovered?
_____________________________________________________________
Answer key on page 30

20. 16
Evaluating a function means replacing the variable in the function, in this
case x, with a value from the function's domain and computing for the result.
To denote that we are evaluating 𝑓 at a for some 𝑎 in the domain of f, we write
𝑓(𝑎).
Check this link for more examples:
https://www.mathsisfun.com/algebra/functions-evaluating.html
LESSON
2 EVALUATING FUNCTIONS
PRE-REQUISITE SKILLS:
You need a good grasp of GEMDAS. GEMDAS is an acronym for the words
Grouping symbols, Exponents, Multiplication, Division, Addition, Subtraction. When
asked to simplify two or more operations in one algebraic/numerical expression, the
order of the letters in GEMDAS indicates what to calculate first, second, third and so
on, until a simplified expression is achieved.
What’s More
Example 1. Evaluate the following functions at 𝑥 = 1.5:
a. 𝑓(𝑥) = 3𝑥 − 2
b. 𝑔(𝑥) = 3𝑥2
− 4𝑥
c. ℎ(𝑥) = √𝑥 + 4
d. 𝑟(𝑥) =
2𝑥+1
𝑥−1
e. 𝑡(𝑥) = ⌊𝑥⌋ + 1 where is the greatest integer function
Solution:
a. 𝑦 = 3𝑥 − 2 = 3(1.5) − 2 = 4.5 − 2 = 2.5
b. 𝑦 = 3𝑥2
− 4𝑥 = 3(1.5)2
− 4(1.5) = 3(2.25) − 6 = 6.75 − 6 = 0.75
c. 𝑦 = √𝑥 + 4 = √1.5 + 4 = √5.5 = 2.34
d. 𝑦 =
2𝑥+1
𝑥−1
=
2(1.5)+1
1.5−1
=
3+1
0.5
=
4
0.5
= 8
e. 𝑦 = ⌊𝑥⌋ + 1 = ⌊1.5⌋ + 1 = 1 + 1 = 2
Example 2.

21. 17
Evaluate the following functions, where f and q are as defined in
Example 1.
a) 𝑓(2𝑥 + 1) b) 𝑔(4𝑥 − 3)
Solution:
a. 𝑓(2𝑥 + 1) = 3(2𝑥 + 1) − 2 = 6𝑥 + 3 − 2 = 𝟔𝒙 + 𝟏
b. 𝑔(4𝑥 − 3) = 3(4𝑥 − 3)2
− 4(4𝑥 − 3)
= 3(16𝑥2
− 24𝑥 + 9) − 16𝑥 + 12
= 48𝑥2
− 72𝑥 + 27 − 16𝑥 + 12
= 48𝑥2
− 88𝑥 + 39
Example 3
Evaluate 𝑓(𝑎 + 𝑏) where 𝑓(𝑥) = 4𝑥2
− 3𝑥 .
Solution.
𝑓(𝑎 + 𝑏) = 4(𝑎 + 𝑏)2
− 3(𝑎 + 𝑏) = 4(𝑎2
+ 2𝑎𝑏 + 𝑏2) − 3𝑎 − 3𝑏
= 4𝑎2
− 3𝑎 + 8𝑎𝑏 − 3𝑏 + 4𝑏2
Example 4
Suppose that 𝑠 (𝑇) is the top speed (in km per hour) of a runner when the
temperature is T degrees Celsius. Explain what the statements 𝑠(15) = 12
and 𝑠(30) = 10 mean.
Solution.
The first equation means that when the temperature is 15°𝐶, then the top
speed of a runner is 12 km per hour. However, when temperature rises
to 30°𝐶, the top speed is reduced to 10 km per hour.
Example 5
The velocity 𝑉 (in m/s) of a ball thrown upward 𝑡 seconds after the ball was
thrown is given by 𝑉(𝑡) = 20 – 9.8𝑡. Calculate 𝑉(0) and 𝑉(1), and explain
what these results mean.
Solution.

22. 18
𝑉(0) = 20 – 9.8(0) = 20 and 𝑉(1) = 20 – 9.8(1) = 10.2. These results
indicate that the initial velocity of the ball is 20 m/s. After 1 second, the ball
is traveling more slowly, at 10.2 m/s.
Activity 2 : IRF- Initial, Revised, Final (revised)
How can challenging problems involving functions be analyzed and
solved?
Initial Answer Revised Answer Final Answer
What’s New
Try to solve the following Exercises.
Activity 2: Check it out
a) Evaluate the following functions at 𝑥 = −3
1. 𝑓(𝑥) = 𝑥3
− 64
2. 𝑔(𝑥) = |𝑥3
− 3𝑥2
+ 3𝑥 − 1|
3. 𝑟(𝑥) = √3 − 2𝑥
4. 𝑞(𝑥) =
3𝑥+1
𝑥2+7𝑥+10
b) Given 𝑓(𝑥) = 𝑥2
− 4𝑥 + 4, solve for:
1. 𝑓(3)
2. 𝑓(𝑥 + 3)
c) A computer shop charges P20.00 per hour (or a fraction of an hour) for
the first two hours and an additional P10.00 per hour for each
succeeding hour. Find how much you would pay if you used one of their
computers for:
1) 40 minutes 2) 3 hours 3) 150 minutes
d) Under certain circumstances, a rumor spreads according to the

23. 19
equation
𝑝(𝑡) =
1
1 + 15(2.1)−0.3𝑡
where 𝑝(𝑡) is the proportion of the population that knows the rumor (𝑡)
days after the rumor started. Find 𝑝(4) and 𝑝(10), and interpret the
results.
What I Learned…
You encountered a lot of concepts related to functions. Now it’s time to pause for
a while and reflect to your learning process by doing the 3-2-1 Chart.
What are the 3 most important things you learned?
What are the two things you are not sure about?
What is 1 thing you want to clarify immediately?

24. 20
LESSON
3 Operations on Functions & Composition of Functions
PRE-REQUISITE SKILLS:
Basic knowledge of algebra is required such as simplifying expressions, factoring
and the like.
Source: https://study.com/academy/lesson/what-is-pemdas-definition-rule-examples.html
Learning Outcome(s): At the end of the lesson, the learner is able to perform
addition, subtraction, multiplication, division, composition of functions, and solve
problems involving functions.
Lesson Outline:
1. Review: Operations on algebraic expressions
2. Addition, subtraction, multiplication, and division of functions
3. Function composition
Example 1. Find the sum of
𝟏
𝟑
and
𝟐
𝟓
Solution. The LCD of the two fractions is 15.
1
3
+
2
5
=
5
15
+
6
15
=
5+6
15
=
11
15
Example 2. Find the sum of
1
𝑥−3
and
2
𝑥−5
Solution. The LCD of the two fractions is (𝑥 − 3)(𝑥 − 5) = 𝑥2
− 8𝑥 + 15
1
𝑥 − 3
+
2
𝑥 − 5
=
1(𝑥 − 5)
𝑥2 − 8𝑥 + 15
+
2(𝑥 − 3)
𝑥2 − 8𝑥 + 15
=
𝑥 − 5 + 2𝑥 − 6
𝑥2 − 8𝑥 + 15
=
3𝑥 − 11
𝑥2 − 8𝑥 + 15
Answer key on page 30
RECALL: Addition and Subtraction
a. Find the least common denominator (LCD) of both fractions.
b. Rewrite the fractions as equivalent fractions with the same LCD.
c. The LCD is the denominator of the resulting fraction.
d. The sum or difference of the numerators is the numerator of the resulting
fraction.

25. 21
Example 3. Find the product of
10
21
and
15
8
.
Solution.
Express the numerators and denominators of the two fractions into
their prime factors. Multiply and simplify out common factors in the
numerator and the denominator to reduce the final answer to lowest
terms.
10
21
●
15
8
=
2 ● 5
3 ● 7
●
3 ● 5
2 ● 4
=
25
28
Example 4. Find the product of
𝑥2−4𝑥−5
𝑥2−3𝑥+2
and
𝑥2−5𝑥+6
𝑥2−3𝑥−10
.
Solution.
Express the numerators and denominators of the two rational
expressions into their prime factors. Multiply and simplify out common
factors in the numerator and the denominator to reduce the final
answer to lowest terms. Note the similarity in the process between this
example and the previous one on fractions.
𝑥2
− 4𝑥 − 5
𝑥2 − 3𝑥 + 2
●
𝑥2
− 5𝑥 + 6
𝑥2 − 3𝑥 − 10
=
(𝑥 + 1)(𝑥 − 5)
(𝑥 − 1)(𝑥 − 2)
●
(𝑥 − 2)(𝑥 − 3)
(𝑥 − 5)(𝑥 + 2)
=
(𝑥 + 1)
(𝑥 − 1)
●
(𝑥 − 3)
(𝑥 + 2)
=
𝑥2
− 2𝑥 − 3
𝑥2 + 𝑥 − 2
RECALL: Multiplication
a. Rewrite the numerator and denominator in terms of its prime factors.
b. Common factors in the numerator and denominator can be simplified as “1”.
c. Multiply the numerators together to get the new numerator.
d. Multiply the denominators together to get the new denominator.

26. 22
Example 5. Divide
2𝑥2+𝑥−6
2𝑥2+7𝑥+5
and
𝑥2−2𝑥−8
2𝑥2−3𝑥−20
Solution:
2𝑥2
+ 𝑥 − 6
2𝑥2 + 7𝑥 + 5
÷
𝑥2
− 2𝑥 − 8
2𝑥2 − 3𝑥 − 20
=
2𝑥2
+ 𝑥 − 6
2𝑥2 + 7𝑥 + 5
●
2𝑥2
− 3𝑥 − 20
𝑥2 − 2𝑥 − 8
=
(2𝑥 − 3)(𝑥 + 2)
(2𝑥 + 5)(𝑥 + 1)
●
(2𝑥 + 5)(𝑥 − 4)
(𝑥 + 2)(𝑥 − 4)
=
(2𝑥 − 3)
(𝑥 + 1)
RECALL: Division
To divide two fractions or rational expressions, multiply the dividend with the
reciprocal of the divisor.
(𝒇 + 𝒈)(𝒙) = 𝒇(𝒙) + 𝒈(𝒙)
(𝒇 − 𝒈)(𝒙) = 𝒇(𝒙) − 𝒈(𝒙)
(𝑓●𝑔)(𝑥) = 𝑓(𝑥)●𝑔(𝑥)
Definition.
Let 𝑓 and 𝑔 be functions.
1. Their sum, denoted by 𝑓 + 𝑔 , is the function denoted by
2. Their difference, denoted by 𝑓 − 𝑔 , is the function denoted by
3. Their product, denoted by 𝑓●𝑔 , is the function denoted by
4. Their quotient, denoted by
𝑓
𝑔
, is the function denoted by
(
𝑓
𝑔
) (𝑥) =
𝑓(𝑥)
𝑔(𝑥)
, excluding the values of x where 𝑔(𝑥) = 0.

27. 23
Use the following functions below for Example 5
 𝒇(𝒙) = 𝒙 + 𝟑
 𝒑(𝒙) = 𝟐𝒙 − 𝟕
 𝒗(𝒙) = 𝒙𝟐
+ 𝟓𝒙 + 𝟒
 𝒈(𝒙) = 𝒙𝟐
+ 𝟐𝒙 − 𝟖
 𝒉(𝒙) =
𝒙+𝟕
𝟐−𝒙
 𝒕(𝒙) =
𝒙+𝟐
𝒙+𝟑
Example 6. Determine the following functions.
a) (𝑣 + 𝑔)(𝑥)
b) (𝑓 ● 𝑝)(𝑥)
c) (𝑓 + ℎ)(𝑥)
d) (𝑝 − 𝑓)(𝑥)
e) (
𝑣
𝑔
) (𝑥)
Solution:
a. (𝑣 + 𝑔)(𝑥) = (x2
+ 5x + 4) + (x2
+ 2x − 8)
= 𝑥2
+ 5𝑥 + 4 + 𝑥2
+ 2𝑥 − 8
= 2𝑥2
+ 7𝑥 − 4
b. (𝑓 ● 𝑝)(𝑥) = (𝑥 + 3)(2𝑥 − 7) = 2𝑥2
− 𝑥 − 21
c. (𝑓 + ℎ)(𝑥) = (𝑥 + 3) +
𝒙+𝟕
𝟐−𝒙
=
(𝑥 + 3)(2 − 𝑥)
2 − 𝑥
+
𝑥 + 7
2 − 𝑥
=
(𝑥 + 3)(2 − 𝑥) + 𝑥 + 7
2 − 𝑥
=
6 − 𝑥 − 𝑥2
+ 𝑥 + 7
2 − 𝑥
=
13 − 𝑥2
2 − 𝑥
=
𝑥2
− 13
𝑥 − 2
d. (𝑝 − 𝑓)(𝑥) = (2𝑥 − 7) − (𝑥 + 3) = 2𝑥 − 7 − 𝑥 − 3 = 𝑥 − 10
e. (
v
g
) (x) =
x2+5x+4
x2+2x−8
=
(x+1)(x+4)
(𝑥−2)(𝑥+4)
=
(x+1)
(𝑥−2)
Applying operations on functions may be quite confusing but as soon as you fully
learn the concept, you can derive strategies to simplify functions easily.
For further understanding on this lesson, watch the video using the link below,

28. 24
https://www.youtube.com/watch?v=lIbAiPUrtvQ
For examples 7 to 10, use the following functions:
𝑓(𝑥) = 2𝑥 + 1 𝑔(𝑥) = √𝑥 + 1 𝑝(𝑥) =
2𝑥+1
𝑥−1
𝑞(𝑥) = 𝑥2
− 2𝑥 + 2 𝐹(𝑥) = ⌊𝑥⌋ + 1
Example 7: Find and simplify 𝑔 ○ 𝑓 (𝑥)
Solution:
𝑔 ○ 𝑓 (𝑥) = 𝑔(2𝑥 + 1) = √2𝑥 + 1 + 1 = √2𝑥 + 2
Example 8: Find and simplify 𝑞 ○ 𝑓 (𝑥)
Solution:
𝑞 ○ 𝑓 (𝑥) = (2𝑥 + 1)2
− 2(2𝑥 + 1) + 2
= 4𝑥2
+ 4𝑥 + 1 − 4𝑥 − 2 + 2
= 4𝑥2
+ 1
Example 9: Find and simplify 𝑓 ○ 𝑝 (𝑥)
Solution:
𝑓 ○ 𝑝 (𝑥) = 2 (
2𝑥 + 1
𝑥 − 1
) + 1
=
(4𝑥 + 2) + (𝑥 − 1)
𝑥 − 1
=
5𝑥 + 1
𝑥 − 1
Example 10: Find and simplify 𝐹 ○ 𝑝 (5)
Solution:
Definition.
Let 𝑓 and 𝑔 be functions.
The composite function denoted by 𝑓 ○ 𝑔 is defined by
𝑓 ○ 𝑔 (𝑥) = 𝑓(𝑔(𝑥)).
The process of obtaining a composite function is called function composition.

29. 25
𝐹 ○ 𝑝 (5) = ⌊
2(5) + 1
5 − 1
⌋ + 1 =
11
4
+ 1 = 2 + 1 = 3
PROBLEMS INVOLVING FUNCTIONS
Example 11
Suppose that 𝑁(𝑥) = 𝑥 denotes the number of shirts sold by a shop, and
the selling price per shirt is given by 𝒑(𝒙) = 𝟐𝟓𝟎 – 𝟓𝒙, for 0 ≤ 𝑥 ≤ 20.
Find (𝑁 ● 𝑝)(𝑥) and describe what it represents.
Solution:
(𝑁 ● 𝑝)(𝑥) = 𝑁(𝑥)●𝑝(𝑥) = 𝑥 (𝟐𝟓𝟎 – 𝟓𝒙) = 𝟐𝟓𝟎𝒙 − 𝟓𝒙𝟐
, 0 ≤ 𝑥 ≤ 20. Since
this function is the product of the quantity sold and the selling price, then
(𝑁 ● 𝑝)(𝑥) represents the revenue earned by the company.
Example 12
A spherical balloon is being inflated. Let 𝑟(𝑡) = 3𝑡 cm represent its radius at
time 𝑡 seconds, and let 𝑔(𝑟) =
4
3
𝜋𝑟3
be the volume of the same balloon if its
radius is 𝑟. Write (𝑔 ○ 𝑟) in terms of 𝑡, and describe what it represents.
Solution:
(𝑔 ○ 𝑟) = 𝑔(𝑟(𝑡) =
4
3
𝜋𝑟(3𝑡)3
=
4
3
𝜋(27𝑡3) = 36𝜋𝑡3
. This
function represents the volume of the balloon at time t
seconds.
What’s More
Activity 3: We Co-Operate
a) Let f and g be defined as 𝑓(𝑥) = 𝑥 − 5 and 𝑔(𝑥) = 𝑥2
− 1 . Find,
1. 𝑓 + 𝑔 4.
𝑓
𝑔
2. 𝑓 − 𝑔 5.
𝑔
𝑓
3. 𝑓●𝑔
b) Let 𝑓(𝑥) = 𝑥2
− 1 and 𝑔(𝑥) =
1
𝑥
, find
1. 𝑓 ○ 𝑔 (𝑥)
2. 𝑔 ○ 𝑓(−1)
3. 𝑓 ○ 𝑓(𝑥)

30. 26
4. 𝑔 ○ 𝑔(5)
c) Evaluate the following composition of functions.
Given :
𝑓(𝑥) = 2𝑥 + 1
𝑔(𝑥) = 5𝑥2
ℎ(𝑥) = 𝑥 + 3
1. (𝑓 ∘ 𝑔)(𝑥)
2. (𝑔 ∘ 𝑓)(𝑥)
3. (ℎ ∘ 𝑔)(𝑥)
4. (𝑓 ∘ ℎ)(𝑥)
d) Suppose that 𝑁(𝑥) = 𝑥 denotes the number of bags sold by a shop, and the
selling price per bag is given by 𝑝(𝑥) = 320 – 8𝑥, for 0 ≤ 𝑥 ≤ 10. Suppose
further that the cost of producing x bags is given by 𝐶(𝑥) = 200𝑥. Find
1. (𝑁 ● 𝑝)(𝑥) and
2. (𝑁 ● 𝑝 – 𝐶)(𝑥).
What do these functions represent?
Application
Let x represent the regular price of a book.
1. Give a function 𝑓 that represents the price of the book if a P100 price
reduction applies.
2. Give a function 𝑔 that represents the price of the book if a 10% discount
applies.
3. Compute (𝑓 ○ 𝑔)(𝑥) and (𝑔 ○ 𝑓)(𝑥). Describe what these mean. Which of
these give a better deal for the customer?
Process questions:
1. What information would help you solve the given problem?
2. What property can be used to solve the problem and why?
3. Show your solution and justification.
4. How can challenging problems involving functions be analyzed and solved?
Answer key on page 31

31. 27
Generalization
You encountered a lot of concepts related to functions. Now it’s time to pause for
a while and reflect to your learning process by doing the 3-2-1 Chart.
Let us summarize…
Key Concepts
 A function is a set of ordered pairs (x,y) such that no two ordered pairs have
the same x-value but different y-values. Using functional notation, we can
write f(x) = y, read as “f of x is equal to y.”
 A function can be presented in the following ways: as a set of ordered pairs,
as a rule or equation, as a table of values, as a mapping diagram (one -to-
one, many-to-one), and through graphs.
 To check whether a graph represents a function, the vertical-line test is
applied.
 A piece-wise function is a function that contains several expressions
depending on restrictions of values the unknown variable will take on in a
certain situation
What are the 3 most important things you learned?
What are the two things you are not sure about?
What is 1 thing you want to clarify immediately?

32. 28
 To evaluate a function means to substitute/replace the variable with a given
value or an expression. f(a) denotes that f will be computed by replacing all
the variables in the functions with a.
 Operations on functions is denoted by the following:
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). .
Their product, denoted by 𝑓  𝑔, is the function denoted by
(𝑓𝑔)(𝑥) = 𝑓(𝑥) (𝑥).
Their quotient, denoted by f ÷g, is the function denoted by
(𝑓 ÷ 𝑔)(𝑥) =
𝑓(𝑥)
𝑔(𝑥)
, excluding the values of x where g(x)=0.
 The composition of the function “ 𝑓 𝑜𝑓 𝑔 ” is defined as follows:
(𝑓  𝑔)(𝑥) = 𝑓(𝑔(𝑥)). This means that 𝑓(𝑥) is composed of the function
𝑔(𝑥). In other words, the variable 𝑥 in 𝑓(𝑥) will take on the value of 𝑔(𝑥).
 In solving composite functions, it is important to apply the GEMDAS principle.
 Real-life problems/scenarios could be represented by functions.
POSTTEST
Let us check how much you have learned about functions.
Direction: Choose the letter of the best answer and write this on your answer sheet.
1. Given 𝑓(𝑥) = 2𝑥 − 5 & 𝑔(𝑥) = 3𝑥 + 4, solve for 𝑔 ○ 𝑓(𝑥).
a. 11 − 6𝑥 c. 6𝑥 − 11
b. 6𝑥2
− 7𝑥 − 20 d. 6𝑥2
− 23𝑥 − 20
2. Given 𝑦 = 3𝑥 + 7, what is 𝑓(−2)?
a. 1 c. -13
b. -1 d. 13
3. The composite function denoted by 𝑓 ○ 𝑔 is defined by.
a. 𝑓 ○ 𝑔(𝑥) = 𝑓(𝑔(𝑥)) c. 𝑓 ○ 𝑔(𝑥) = 𝑓(𝑥)●𝑔(𝑥)
b. 𝑓 ○ 𝑔(𝑥) = 𝑔(𝑓(𝑥)) d. 𝑓 ○ 𝑔(𝑥) = 𝑔(𝑥)●𝑓(𝑥)
4. It is a set of ordered pairs (𝑥, 𝑦) such that no two ordered pairs have the same x-
value but different y-values?

33. 29
a. relation c. domain
b. function d. range
5. What is the domain of the equation, 𝑦 = 3𝑥2
− 4𝑥?
a. {𝒙: 𝒙 ∈ 𝑹, 𝒙 < −𝟏} c. {𝒙: 𝒙 ∈ 𝑹}
b. {𝒙: 𝒙 ∈ 𝑹, 𝒙 ≠ 𝟏} d. {𝒙: 𝒙 ∈ 𝑹, 𝒙 ≥ 𝟒}
6. Given 𝑓(𝑥) = 2𝑥 − 5 & 𝑔(𝑥) = 3𝑥 + 4, solve for 𝑓●𝑔(𝑥)
a. 6𝑥2
+ 23𝑥 − 20 c. 6𝑥2
− 20
b. 6𝑥2
− 23𝑥 − 20 d. 6𝑥2
− 7𝑥 − 20
7. If 𝑓(𝑥) = 𝑥 + 7 & 𝑔(𝑥) = 2𝑥 − 3, what is 𝑓 − 𝑔(𝑥)
a. −𝑥 + 4 c. 𝑥 − 4
b. 10 − 𝑥 d. 10 + 3𝑥
8. To divide two fractions or rational expressions, multiply the dividend with the
________ of the divisor.
a. reciprocal c. abscissa
b. addend d. Theorem
9. The ___ of a relation is the set of all possible values that the variable x can take.
a. domain c. equation
b. range d. function
10.Which of the following set of ordered pairs in NOT a function?
a. (1,2), (2,3), (3,4), (4,5) c. (1, 1), (2, 2), (3, 3), (4, 4)
b. (1,2), (1,3), (3,6), (4,8 d. (3, 2), (4, 2), (5, 2), (6, 2)
11.A graph represents a function if and only if each vertical line intersects the graph
at most _____.
a. once c. twice
b. thrice d. all of the these
12.What is the domain of the function 𝑦 = √𝑥 − 5 ?
a. {𝑥: 𝑥 ∈ 𝑅, 𝑥 ≥ −5} c. {𝑥: 𝑥 ∈ 𝑅, 𝑥 ≥ 5}
b. {𝑥: 𝑥 ∈ 𝑅, 𝑥 ≤ −5} d. {𝑥: 𝑥 ∈ 𝑅, 𝑥 ≤ 5}
13.The composite function denoted by 𝑓 ○ 𝑔 is defined by ___________.
a. 𝑓 ○ 𝑔 (𝑥) = 𝑔(𝑥) c. 𝑓 ○ 𝑔 (𝑥) = 𝑔(𝑓(𝑥))
b. 𝑓 ○ 𝑔 (𝑥) = 𝑓(𝑔(𝑥)) d. 𝑓 ○ 𝑔 (𝑥) = 𝑓(𝑥)
14.Given 𝑓(𝑥) = 4𝑥2
− 3𝑥, what is 𝑓(−2)?
a. −22 c. 22
b. −10 d. 10
15.The quotient, denoted by
𝑓
𝑔
, is the function denoted by (
𝑓
𝑔
) (𝑥) =
𝑓(𝑥)
𝑔(𝑥)
,

34. 30
excluding the values of x where 𝑔(𝑥) = _________.
a. 0 c. 1
b. Both a and c d. None of these
ANSWER KEY
PRETEST
1) C 6) D 11) A
2) A 7) B 12) C
3) A 8) A 13) B
4) B 9) A 14) C
5) C 10) B 15) A
Activity 1:
1. All real numbers, except 2 & 4
2. Solution.
C. All diagrams, except for C, represent a function
3. {𝑋: 𝑋 ∈ 𝑅, 𝑋 < 7}
4. 𝑆(𝐸) = 600𝑛
5. 𝐶(𝑟) = 40 + 3.50𝑑
Activity 2
a. Item
5. 𝑓(𝑥) = 𝑥3
− 64 = (−3)3
− 64 = −27 − 64 = 91
6. 𝑔(𝑥) = |𝑥3
− 3𝑥2
+ 3𝑥 − 1| = 64
7. 𝑟(𝑥) = √3 − 2𝑥 = 3
8. 𝑞(𝑥) =
3𝑥+1
𝑥2+7𝑥+10
= 4
b. Given 𝑓(𝑥) = 𝑥2
− 4𝑥 + 4, solve for:
3. 𝑓(3) = 1
4. 𝑓(𝑥 + 3) = 𝑥2
+ 2𝑥 + 1
c. A computer shop charges P20.00 per hour (or a fraction of an hour) for the
first two hours and an additional P10.00 per hour for each succeeding hour.
Find how much you would pay if you used one of their computers for:
2) 40 minutes =
20
3
= 6.67 Pesos

35. 31
3) 3 hours = 30 Pesos
4) 150 minutes = 25 Pesos
Activity 3
e) Let f and g be defined as 𝑓(𝑥) = 𝑥 − 5 and 𝑔(𝑥) = 𝑥2
− 1 . Find,
1. 𝑓 + 𝑔 = 𝑥2
+ 𝑥 − 6 4.
𝑓
𝑔
=
𝑥−5
𝑥2−1
4. 𝑓 − 𝑔 = −𝑥2
+ 𝑥 − 4 5.
𝑔
𝑓
=
𝑥2−1
𝑥−5
5. 𝑓●𝑔 = 𝑥3
− 5𝑥2
− 𝑥 + 5
f) Let 𝑓(𝑥) = 𝑥2
− 1 and 𝑔(𝑥) =
1
𝑥
, find
5. 𝑓 ○ 𝑔 (𝑥) =
1−𝑥2
𝑥2
6. 𝑔 ○ 𝑓(−1) =
1
2
7. 𝑓 ○ 𝑓(𝑥) = 𝑥4
− 2𝑥2
8. 𝑔 ○ 𝑔(5) = 5
g) (𝑥) = 2𝑥 + 1 ; 𝑔(𝑥) = 5𝑥2
; ℎ(𝑥) = 𝑥 + 3
1. ( 𝑓 ∘ 𝑔 ) ( 𝑥 ) = 2( 5𝑥2
) + 1
= 10𝑥2
+ 1
2. ( 𝑔 ∘ 𝑓 ) ( 𝑥 ) = 5(2𝑥 + 1 )2
= 20𝑥2
+ 20𝑥 + 5
3. ( ℎ ∘ 𝑔 ) ( 𝑥 ) = (5𝑥2
) + 3
4. ( 𝑓 ∘ ℎ ) ( 𝑥 ) = 2 (𝑥 + 3) + 1
= 2𝑥 + 7
h) Suppose that 𝑁(𝑥) = 𝑥 denotes the number of bags sold by a shop, and the selling
price per bag is given by 𝑝(𝑥) = 320 – 8𝑥, for 0 ≤ 𝑥 ≤ 10. Suppose further that
the cost of producing x bags is given by 𝐶(𝑥) = 200𝑥. Find
3. (𝑁 ● 𝑝)(𝑥) = 320𝑥 − 8𝑥2
– Gross value
4. (𝑁 ● 𝑝 – 𝐶)(𝑥) = 120𝑥 − 8𝑥2
– Net value
POSTTEST
1) C 6) D 11) A
2) A 7) B 12) C
3) A 8) A 13) B
4) B 9) A 14) C
5) C 10) B 15) A

36. 32
REFERENCES
General Mathematics pg. 1-20
Department of Education Teachers Materials
Math is Fun
https://www.mathsisfun.com/algebra/functions-evaluating.html
Ronie Banan, June 30, 2018
https://www.youtube.com/watch?v=lIbAiPUrtvQ
MathEase, September 1, 2014
https://www.youtube.com/watch?v=tAoe4xjUZQk
For inquiries and feedback, please write or call:
Department of Education – Bureau of Learning Resources (DepEd-BLR)
Division of Misamis Oriental
Don Apolinar Velez St., Cagayan de Oro City 9000
Contact Number: 0917 899 2245
Misamis.oriental@deped.gov.ph