SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
MODULE IN
GENERAL MATHEMATICS
COURSE DESCRIPTION:
At the end of the course, the students must know how to solve problems
involving rational, exponential and logarithmic functions; to solve business-related problems; and to apply logic to
real-life situations.
GRADE AND SECTION: Grade 11
CLASS SCHEDULE:
CLASS ONLINE ACCOUNT:
PhilHealth
Accredited
PAASCU
Accredited
SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
PART 1
MAIN TOPIC/ SUB TOPIC
Learning Module No.
Quarter
Week No.
Date
1
1
1
Functions
BRIEF INTRODUCTION OF THE TOPIC
A function is a relation between dependent and independent variable/s wherein for every value of the
independent variable where the function is defined, there exist a unique or single value of the dependent
variable.
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).
MOST ESSENTIAL LEARNING COMPETENCIES
At the end of the lesson, the students should be able to:
1) represent real-life situations using functions; and
2) evaluate a function
VALUE/BEHAVIORAL STANDARDS
● Commitment to Vincentian Excellence
● Simplicity
INSTRUCTIONAL MATERIALS
● Articles
● Handouts and Worksheets
● Audio-Visual Media
LEARNING RESOURCES
Books
● Apolinario,P. L., Infante,F. N. M., Garces, L. P. D. M., Sin, J. L. M., Tresvalles, R. M., and Verzosa, D.
M. B. (2016). Mathematics Learner’s Module. Pasig City, Philippines: Department of Education,
Lexicon Press Inc.
● Apolinario,P. L., Infante,F. N. M., Garces, L. P. D. M., Sin, J. L. M., Tresvalles, R. M., and Verzosa, D.
M. B. (2016). Teaching Guide: General Mathematics. Quezon City, Philippines: Department of
Education, EC-TEC Commercial.
PhilHealth
Accredited
PAASCU
Accredited
SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
● Magcuyao, D. A. H. (2016). General Mathematics. JFS Publishing Services. Pasay City.
Electronic Sources
● https://courses.lumenlearning.com/intermediatealgebra/chapter/read-define-and-write-piecewisefunctions/
● Khan
Academy.
Introduction
To
Piecewise
Functions.
2020.
https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:absolute-value-piecewisefunctions/x2f8bb11595b61c86:piecewise-functions/v/piecewise-function-example (accessed July 18,
2020).
● Courses.lumenlearning.com. Define And Write Piecewise Functions | Intermediate Algebra. 2020.
https://courses.lumenlearning.com/intermediatealgebra/chapter/read-define-and-write-piecewisefunctions/ (accessed July 18, 2020).
LEARNING MODALITIES
�
�
�
Modular Learning – Modules are provided to support immediate learning in the absence of face-face
classroom.
Online Delivery/Remote Delivery – While using modules, synchronous or real-time communication
platforms shall still be utilized in order to support student learning. Available applications (such as
Google Classroom, Facebook page and among others) will be used for interactive classroom
discussion, chat platform and online submissions.
Asynchronous Delivery – Pre-taken teaching videos (or any available means) will also be done to
help learners who might experience difficulties with real-time class interactions.
PhilHealth
Accredited
PAASCU
Accredited
SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
PART 2
PRE – ASSESSMENT
LEARNING EXERCISE
Part 1. Identify if the following scenarios represent a FUNCTION or NOT. Write your answers on the
worksheet uploaded on our Google Classroom.
1.
2.
3.
4.
5.
PhilHealth
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PAASCU
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SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
Part 2. Given a function --- C(x) = 40x. Evaluate the following:
6. x=3
7. x=-2
8. C(0)
LEARNING OBJECTIVES
At the end of this lesson, (the learners)you are expected to:
a) Functions as Models
✔ define and differentiate a relation and function;
✔ represent a relation and function by using ordered pair, mapping diagram, graph
and equation; and
✔ represents real-life situations using functions, including piece-wise functions.
b) Evaluation of Function
✔ identify if an equation is a function or not; and
✔ evaluates a function.
DISCUSSION OF THE CONTENT OF THE LESSON/TOPIC
Lesson 1. Functions as Models
● Definition of Relation and Function
Relation:
● 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).
Function:
● 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 xvalue, 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 can say that f(1) = 2.
EXAMPLE 1
Which of the following relations are functions?
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2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
Solution
The relations f and h are functions because no two ordered pairs have the same x-value but different yvalues. Meanwhile, g is not a function because (1,3) and (1,4) are ordered pairs with the same x-value but
different y-values.
RELATION IN MAPPING DIAGRAMS
● 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.
EXAMPLE 2
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=7 corresponds to y = 11 or 13. Similarly, x=2 corresponds to both y=17 or 19.
RELATION IN A GRAPH
● 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.
● The Vertical Line Test
o A graph represents a function if and only if each vertical line intersects the graph at most once.
PhilHealth
Accredited
PAASCU
Accredited
SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
EXAMPLE 3
Which of the following can be graphs of functions?
Solution
Graphs a.), b.), c.) are graphs of functions while d.) and e.) are not because they do not pass the vertical line
test.
EXAMPLE 4 : Relation In An Equation
✔ Which of the following represents a function?
Solution
Both a and b are functions while c is not a because the y-variable in the equation has an exponent greater
than 1
● Functions as representations of real-life situations
● Functions are often used to model real-life situations. Identifying an appropriate functional
model will lead to a better understanding of various phenomena.
PhilHealth
Accredited
PAASCU
Accredited
SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
EXAMPLE 5
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.
EXAMPLE 6
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 the person went to work.
Solution
Since the person is earning 600 pesos a day, then the cost function is S(n) = 600n.
EXAMPLE 7
If height (H) is a function of age (a), give a function H that can represent the height of a person in a age, if
every year the height is added by 2 inches.
Solution
Since every year the height is added by 2 inches, then the height function is �(�) = � + �
Lesson 2. Evaluating a Function
● When the function f(x) takes the value of an independent variable, the function is said to be evaluated
at the given point. When evaluating a function at the given point, the following procedure may be
considered:
1. Identify the independent variable and replace each independent variable by a pair of
parentheses.
2. Plug in the given value of the independent variable to the equation and simplify the
expression using the PEMDAS rule.
● Ensure that there is no violation of rules in algebra like division by 0, and negative radicand for even
indexed radicals.
o LAW OF SUBSTITUTION
If a+x=b and x=c, then a+c=b
PhilHealth
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SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
EXAMPLE 1
Evaluate the following functions given x=1.5:
Solution
EXAMPLE 2
Find g(-4) and r(1) where g and r are defined below:
Solution
It is NOT possible because -4 is not in the domain of g(x) and 1 is not in the domain of r(x).
Take note:
● The domain of a relation/function is the set of all possible values that the variable x can take.
PhilHealth
Accredited
PAASCU
Accredited
SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
EXAMPLE 3
Evaluate the following functions where f and q are as defined in EXAMPLE 1.
Solution
--- Substitute x with (3x-1)
--- Distributive Property of Multiplication
--- Combine Like terms
--- Substitute x with (2x+3)
--- Distributive Property of Multiplication
--- Combine Like terms
LEARNING ACTIVITIES
E-LINKS
Hone your skills more by visiting the following sites below:
Audio-visual Presentation:
● https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:eva
luating-functions/v/understanding-function-notation-example-1
PhilHealth
Accredited
PAASCU
Accredited
SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
Activity
We said that for a relation to become a function, the value of the domain must correspond to a single value
of the range. Let’s read a short conversation and determine if it can be classified as function or not.
June and Mae are in a long-time relationship until June realized that he wants to marry Mae.
From the above conversation, do you think it can be classified as function? State the reason/s why the above
scenario is a function or not.
PhilHealth
Accredited
PAASCU
Accredited
SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
EVALUATION /POST ASSESSMENT
LEARNING TASK
The Post-assessment activity will be posted in the Google Classroom, Classwork Tab.
1. Is the relation {(0,0), (1,1), (2,4), (3,9), ... (n, n2), ...} a function?
2. Which of the following diagram represents a relation that is NOT a function?
PhilHealth
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PAASCU
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SAN JUAN DE DIOS EDUCATIONAL FOUNDATION, INC.
2772-2774 Roxas Boulevard, Pasay City 1300 Philippines
Hospital - Tel. Nos.: 8-831-9731/36; 8-831-5641/49 Website: www.sanjuandedios.org
College - Tel. Nos.: 8-551-2756; 8-551-2763
Website: www.sjdefi.edu.ph
Senior High School Department
SUBJECT: General Mathematics
Semester: 1st S.Y. 2021-2022
3. Can the graph of a circle be considered a function? Why or Why Not?
4. Create the piecewise function for the scenario:
The cost of hiring catering services for a party is 150 pesos per head for 20 person or less, 130 pesos
per head for 21 to 50 persons, and 110 pesos per head for 51 to 100 persons. For more than 100 people, the
cost is at 100 pesos per head. Represent the total cost as a piecewise function of the number of attendees x of
the party.
5. Given f(x) = x2-2, find the following values:
a) f(0)
d) f(π)
b) f(3)
e) f(x+1)
c) f(-1)
g) f(3x)
6. Mark started selling snacks in a nearby school. In a day, he spends 200 pesos for the store rent and 25
pesos for each snack item he prepares. His expenses in a single day can be expressed as the function C(x)
=____(a)__, where x is the number of items and C(x) is his daily expenses in peso. (b) How much will be his
expenses if he prepares 100 snack items? (c) 150 snack items?
Prepared & Enhanced by:
SONIA RUTH RUBY T. RUBIO, LPT
SHS Faculty, Math and Sciences
Checked by:
DR. CAROLYNE L. HIZON, LPT
Academic Coordinator, Senior High School Department
Approved by:
DR. APOLONIA P. REYNOSO, LPT
Principal, Senior High School Department
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