Functions as Models
Learner's Module in General Mathematics
Quarter 1 ● Module 1 ● Week 1
JOPIE B. FERRER
Developer
Department of Education • Cordillera Administrative Region
NAME:________________________ GRADE AND SECTION ________________
TEACHER: ____________________ SCORE _____________________________
Republic of the Philippines
DEPARTMENT OF EDUCATION
Cordillera Administrative Region
SCHOOLS DIVISION OF BAGUIO CITY
No. 82 Military Cut-off, Baguio City
Published by:
DepEd Schools Division of Baguio City
Curriculum Implementation Division
Learning Resource Management and Development System
COPYRIGHT NOTICE
2020
Section 9 of Presidential Decree No. 49 provides:
“No copyright shall subsist in any work of the Government of the Philippines.
However, prior approval of the government agency of office wherein the work is
created shall be necessary for exploitation of such work for profit.”
This material has been developed for the implementation of K-12 Curriculum through
the DepEd Schools Division of Baguio City – Curriculum Implementation Division
(CID). It can be reproduced for educational purposes and the source must be
acknowledged. Derivatives of the work including creating an edited version, an
enhancement or a supplementary work are permitted provided all original work is
acknowledged and the copyright is attributed. No work may be derived from this
material for commercial purposes and profit.
ii
What I Need to Know
Hello learner! This module was designed and written with you in mind.
Primarily, its scope is to represent real-life situations using functions, including
piecewise functions and to evaluate a function.
While going through this module, you are expected to:
1.
2.
3.
4.
5.
define functions and relations;
distinguish between functions and relations;
identify the different ways of determining a function;
describe real-life situations using piecewise functions by choice; and
determine the value of something in a thoughtful way
Now, here is an outline of the different parts of your learning module. The
descriptions will guide you on what to expect on each part of the module.
Icon
Label
Description
What I need to know
This states the learning objectives that you need to
achieve as you study this module.
What I know
This is to check what you already know about the
lesson on this module. If you answered all the
questions here correctly, then you may skip
studying this module.
What’s In
This connects the current lesson with a topic or
concept necessary to your understanding.
What’s New
This introduces the lesson to be tackled through an
activity.
What’s In it
This contains a brief discussion of the learning
module lesson. Think of it as the lecture section of
the lesson.
What’s More
These are activities to check your understanding
and to apply what you have learned from the
lesson.
What I have Learned
This generalizes the essential ideas tackled from
this module.
What I Can Do
This is a real-life application of what you have
learned.
Post-Assessment
This is an evaluation of what you have learned from
this learning material.
Additional Activity
This is an activity that will strengthen and fortify
your knowledge about the lesson.
2
What I Know
If you answer all the test items correctly in this pretest, then you may skip
studying this learning material and proceed to the next learning module.
I.
TRUE OR FALSE. Write the word TRUE if the given statement is correct and
FALSE if otherwise. (5 points)
__________ 1. A function from to is a set of ordered pair
such that to
each
, there corresponds a unique
.
__________ 2. All functions are relations.
__________ 3. Evaluating a function means to find the value of the output f (x) .
__________ 4. Horizontal line test is used to determine if a graph represents a
function.
__________ 5. A relation is a function defined by two or more formulas on
different parts of its domain.
II. IDENTIFICATION. Determine whether each of the following items is a function or
simply a relation. Write FUNCTION if can be a function and RELATION if only a
relation on the space provided. (5 points)
2
1. _____________________________
4. _____________________________
2. _____________________________
A
1
B
2
C
3
5. _____________________________
3. _____________________________
3
III. MULTIPLE CHOICE. Choose the best answer. Write the letter of your choice in
CAPITAL LETTERS on the space provided for. (5 points)
_____ 1. Which of the following is a function?
A.
B.
C.
D.
_____ 2. Which set considers all possible values of ?
A. Domain
B. Range
C. Function
D. Relation
_____ 3. Evaluate
A. 8
B. 1
D. 6
_____ 4. Given
A.
B.
at
C. 25
C.
, what is the range?
D.
_____ 5. Which of the following graphs is an example of a piecewise function?
A.
C.
B.
D.
4
What’s In
There are many situations wherein a variable depends on another variable. For
example, the distance travelled by a car depends on its rate, the savings of the
government depends on its expenditures, the area of a circle depends on the length
of its diameter, and so on. These relationships are mathematically described by
functions. Many real-life situations and problems can be represented and solved by
mathematical models such as functions.
The concept of sets is also considered in this topic. Remember that “set” in
mathematical concept refers to the collection of things or objects with a common
characteristic. An example of set is the group chat in the messenger that you belong
to. In this group chat, there are members with a common goal or purpose.
What’s New
Activity : I Got This!
Write two (2) things or objects that belong to the given group. Write your
answer in column B.
A
B
Even numbers
Indigenous groups in the Philippines
Subjects when you were in Grade 10
or in the ALS Program
Primary colors
Provinces in Cordillera Administrative
Region
What’s In It
FUNCTIONS AND RELATIONS
Definition of Terms

Relation – a relation from t
is a set of ordered pairs
that to each
, there corresponds at least one
.
5
such

Function – a function from to is a set of ordered pairs
that to each
, there corresponds a unique or exactly one
such
.
Note: The symbol is read as “element”. This also refers to the member of
the given set. In addition to, the small letter is the element or member of set
written in capital letter. In naming of a set, capital letters are used. This
goes to
and
from the definition of relation and function. The first
component in the ordered pair is and the second component is .

Domain – is the set of first coordinates in the ordered pairs. It is the set
of all possible values of . In this module, “D” is used to represent the
domain of the given set.

Range – is the set of second coordinates in the ordered pairs. It is the
set of all possible values of . In this module, “R” is used to represent
the range of the given set.
Three Things Necessary to Form a Relation
A. Non – empty set X
 this set also refers to the domain of the given relation
B. Non – empty set Y
 this set is not necessarily the range of the given relation. There
are instances that not all in the second set are mapped from the
first set. In this case, the range is just a subset of the second
set.
C. The pairing or correspondence or mapping of the two sets
 this pertains to the relationship existing between of and
Ways to Describe a Relation
A. Listing of ordered pairs
Example:
D. Graph
Example:
B. Arrow diagram
Example:
X
Y
1
a
2
b
3
c
E. Equation
Examples:
C. Table
Example:
NOTE: All functions are relations but
not all relations are functions.
6
Ways of Determining Whether a Relation is a Function or Not
A. Listing of ordered pairs
 If there are two or more ordered pairs with the same first
component, then the relation is not a function.
Examples:
1.
The first components of set P are 1, 2, 3, and 16. Thus, there is no
same first component in the ordered pairs. This is a function. The
domain is
and the range is
. Since 1 is
repeated as second component, just write it once in the set of
range.
2.
The first components of set T are 1, 1, 2, and 3. Thus, there is a
repetition of 1 in the ordered pairs. This is not a function. It is just
merely a relation. The domain is
and the range is
. Since 1 is repeated as first component, just write it
once in the set of domain.
B. Arrow diagram
 If the first set is mapped or paired to two or more in the second
set, then it is not a function.
Examples:
X
Y
X
1.
Y
2.
1
a
1
a
2
b
2
b
3
c
3
c
Every element in set X is
paired to at least one and
exactly one element in set Y.
Therefore, it is both a relation
and function. The domain is
and the range is
.
A
Every element in set X is
paired to at least one and
exactly one element in the
second set. Therefore, it is
both
a
relation
and
function. The domain is
and the range
is
.
B
3.
A
B
4.
1
a
1
a
2
b
2
b
3
c
3
c
7
Every element in set A is
paired to at least one element
in set B. Therefore, it is only a
relation.
The
domain
is
and the range is
.
C. Table
 If the two or more
function.
Not all in set A is paired to
at least one or exactly one
element
in
set
B.
Therefore, it is not a
relation nor a function.
(Always refer to the
definition).
-values are the same, then it is not a
Examples:
1.
2.
There is no repetition of values, therefore, it is both a
relation and function. The
domain is
and the range is R
.
There is a repetition of values, therefore, it is just
simply a relation. The domain
is
and the
range is R
D. Equation
 If the exponent of the dependent variable is an odd integer,
then it is a function.
 If the first component is not constant, then it is a function.
Examples:
The exponent of y is 1. It is both relation
and function.
The exponent of y is 2. It is just a relation.
The first component x is constant. It is just a
relation.
The first component x is constant. It is just a
relation.
E. Graph
 Vertical line test is used to test if the graph is a function or not. If
it intersects the graph at more than one point, then it is not a
function. (In the given examples, the broken lines are the vertical
lines for testing).
8
Examples:
Every vertical line in the graph, there is only one point of intersection in
the graph. This concludes that both graphs above are functions.
The vertical line intersects the graph at two points. This concludes that
both graphs above are just simply relations.
PIECEWISE FUNCTIONS
Sometimes a function is defined by different formulas on different parts of its
domain. This function is called a piecewise function. It is also known as piecewisedefined function. In this discussion, is being replaced with
or can be denoted
by other symbols or letters
Well-known Examples of Piecewise Functions
1. Absolute Value Function
| | is defined as
{
Examples:
𝑦
𝑥
𝑦
𝑥
9
| |
| |
| |
| |
|
|
|
|
|
|
|
|
2. Greatest Integer Function
⌊ ⌋ is defined as ⌊ ⌋ the largest integer that is less than or
equal to . It is also known as Floor Function. The deleted point (○)
denotes discontinuity (not included) in the graph while the solid point
(●) signifies continuity (included) in the graph.
Examples:
The number line will help you on this.
Locate the number given in the
number line and look at the integer
on its left. That is the answer. If the
number given is already an integer,
then that is the answer.
⌊
⌊
⌋
⌋
⌊
⌋
⌊
⌋
⌊ ⌋
⌊
⌋
⌊ ⌋
3. Unit Step Function
This function is also known as Heaviside Step Function named after
Oliver Heaviside (1850–1925). The function is defined as:
{
Examples:
1.
2.
3.
4.
If
If
If
If
, then
, then
, then
, then
Note: This will be discussed further
on the next topic on Evaluation of
Functions.
What’s More
Activity 1: Find Me!
How large is the value of the output in the function
if we input
? The amount of space that a spherical container can hold, if the radius of the
10
container can hold, if the radius of the container is 12 cm, can be calculated by using
the formula
. What is the volume of the container? In this activity, we shall
learn how to evaluate functions.
Evaluate the following algebraic expressions, given that
. Show your solution.
and
Example:
Copy the expression.
Substitute the values for each variable.
Multiply the first term.
Add the two terms.
1.
2.
3.
4.
Activity 2: Do Not Guesstimate!
To evaluate a function means to find the value of the output
input . Study the given examples.
Example 1: If
, evaluate at
.
Copy the given function.
Replace with .
Apply PEMDAS rule.
Multiply and .
Combine the terms.
Example 2: Given
, find
Copy the given function.
Replace
Divide.
11
with .
, given the
Example 3: Given
, find
Copy the function.
Replace x with x+1.
Apply the FOIL method.
Combine similar terms.
Now, it’s your turn to evaluate! Always remember that to evaluate is just to
substitute the value of in the given function. Algebraic process will just follow.
Show your solution if necessary.
1. Given h
.
√
2. What is
|
, evaluate at
3. If
in
|?
, evaluate
What I Have Learned
Activity: Now, Decide!
Determine whether each of the following items is a function or simply a relation. Write
F for function and R for relation on the blank provided for.
_____ 1.
_____ 2.
_____ 3.
_____ 4.
_____ 5.
_____ 6.
_____7
.
.
(Grade)
(Subject)
92
92
85
Math
Science
English
12
87
90
Filipino EsP
89
93
93
TLE
MAPEH
AP
.
X
Y
Tuba
Lagawe
Benguet
_____ 8.
Mt. Province
Ifugao
Baguio City
Sagada
_____ 9.
_____ 10.
What I Can Do
Activity: Into Pieces
The process of evaluating piecewise functions is similar to evaluating ordinary
functions, except for the fact that we need to choose the piece or chunk of the
function to be used for the given value of . We can do this by examining the
appropriate interval where the input or the value of belongs.
The process of hiring a catering service to serve food for a party is ₱150 per
head for 20 persons or less, ₱130 per head for 21 to 50 persons, and ₱110 per head
for 51 to 100 persons. For 100 or more persons, the cost is at ₱100 per head. The
piecewise defined function of the number of attendees of the party is given by:
{
Example: How much is to be paid if there are 10 attendees in the party?
Since
and can be found in the interval
we use
.
Copy the expression from the function and equate to
Replace x with 10.
Multiply.
.
Now, it’s your turn to show what you can do. What is the cost of the party if
there are 25 attendees? 110 attendees?
1.
2.
13
Assessment
I.
TRUE OR FALSE. Write the word TRUE if the given statement is correct and
FALSE if otherwise. (5 points)
__________ 1. A function from to is a set of ordered pair
such that to
each
, there corresponds a unique
.
__________ 2. All functions are relations.
__________ 3. Evaluating a function means to find the value of the output f (x) .
__________ 4. Horizontal line test is used to determine if a graph represents a
function.
__________ 5. A relation is a function defined by two or more formulas on
different parts of its domain.
II. IDENTIFICATION. Determine whether each of the following items is a function or
simply a relation. Write FUNCTION if can be a function and RELATION if only a
relation on the space provided.
2
1. _____________________________
4. _____________________________
2. _____________________________
A
1
B
2
C
3
5. _____________________________
3. _____________________________
14
III. MULTIPLE CHOICE. Choose the best answer. Write the letter of your choice in
CAPITAL LETTERS on the space provided. (5 points)
_____ 1. Which of the following is a function?
A.
B.
C.
D.
_____ 2. Which set considers all possible values of ?
A. Domain
B. Range
C. Function
D. Relation
_____ 3. Evaluate
A. 8
B. 1
D. 6
_____ 4. Given
A.
B.
at
C. 25
, what is the range?
C.
D.
_____ 5. Which of the following graphs is an example of a piecewise function?
A.
C.
B.
D.
15
Additional Activity
Activity: Tax Rates in the Philippines
Based on the existing tax structure, you are to define a piecewise function that
models this situation. You may give the function through (a) equation, (b) table
showing some values, or (c) graph.
Tax Rate Table:
Source: https://www.pinterest.ph/pin/722898177661051772/
16
I.
II.
WHAT I HAVE
LEARNED
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
F
R
F
R
R
F
R
R
F
R
WHAT I CAN DO
1.
2.
₱ 3, 250
₱ 11, 000
I.
II.
III.
ASSESSMENT
1. TRUE
2. TRUE
3. TRUE
4. FALSE
5. FALSE
1. FUNCTION
2. FUNCTION
3. FUNCTION
4. RELATION
5. FUNCTION
1. B
2. A
3. C
4. D
5. B
WHAT’S NEW
Answers may vary
WHAT I KNOW
III.
17
1. TRUE
2. TRUE
3. TRUE
4. FALSE
5. FALSE
1. FUNCTION
2. FUNCTION
3. FUNCTION
4. RELATION
5. FUNCTION
1. B
2. A
3. C
4. D
5. B
Examples:
1. 0, 2, 4, 6, 8, 10
2. Isneg, Kankanaey, Ibaloi,
Kalanguya, Ifugao, Bontoc,
Isneg, Tinggian, Iyaplay
Math, Science, Filipino, English
Red, Blue, Yellow
Benguet, Kalinga, Abra, Mt.
Province, Ifugao, Apayao,
3.
4.
5.
ADDITIONAL ACTIVITY
Answers may vary
WHAT’S MORE
Activity 1:
1. 27
2. 24
3. 16
4. -116
Activity 2:
1. ±3
2. 25
2
3. 4x + 3x + 2
ANSWER KEY
REFERENCES
Verzosa, Debbie Marie B., et.al. Teaching Guide for Senior High School – General
Mathematics. Quezon City: Commission on Higher Education, 2016.
Belecina, Rene R., et.al. General Mathematics. Quezon City: Brilliant Creations
Publishing, Inc., 2016.
“Train Law (2020) Income Tax Tables in the Philippines – Pinoy Money Talk.”
Pinterest. https://www.pinterest.ph/pin/722898177661051772/
18
For inquiries or feedback, please write or call:
Department of Education – Schools Division of Baguio City
No. 82 Military Cut-off Road, Baguio City
Telefax: 422-4326 / 422-7819
Email Address: depedbaguiocity@gmail.com
Social Media: facebook.com/DepEdTayoBaguioCity