Von Christopher G. Chua, LPT, MST
[Education Program Supervisor,
DepEd, Eastern Samar Division]
DOLORES NATIONAL SENIOR HIGH SCHOOL
Dolores, Eastern Samar, Philippines
[SY 2016-17]
General
Mathematics
Teaching Guides
General Mathematics Teaching Guides
•••
General Mathematics
Teaching Guides
Von Christopher G. Chua, LPT, MST
[Education Program Supervisor,
DepEd, Eastern Samar Division]
Introduction
The teaching guides included in this document have been
designed for teachers who have been assigned to teach the first
core Mathematics subject in the senior high school, General
Mathematics. It has been patterned with the curriculum guide
mandated by the Department of Education with each plan
explicitly stating the content, the content standards, the
performance standards, and the learning competencies together
with their distinct codes. The codes follow a format such as M11GMIa-1 where the first letter stands for Mathematics, 11 for the grade
level, GM for General Mathematics as course, the roman numeral I
for the quarter and in this case the first, the lowercase letter for the
week where a is for the first week, and finally the last Arabic number
is the unique code for each learning competency.
Instead of creating the guides on a daily basis, the author has
decided to adapt a weekly frame to ascertain that all tasks in the
week are coherent and promote continuity of learning. The
procedures, however, are detailed per unit of instructional time, that
is, one hour sessions with four sessions per instructional week.
The guides follow the general format of a lesson plan and include
five major parts: the objectives and the learning competencies, the
subject matter together with the references and the essential ideas;
the daily instructional procedure; student assessment complete with
the required rubrics; and the assignment.
These guides have been designed for ease of use and to provide
other teachers with ideas for activities. The author takes into
account both the availability of learning materials that are easily
accessible even to teachers in the most remote areas of the
country and the instructional innovations provided by today’s
technology. By elucidating a variety of activities, teachers may opt
to choose the more convenient and appropriate for their learning
environment.
Since this is a work-in-progress, the author also made sure to include
important notes after the actual implementation of each teaching
guide. This serves as evaluation of the effectiveness of the material
and will be used to further improve the guides constructed. It is
therefore highly recommended that teachers who have opted to
use these guides should contact the author to contribute to the
development of these teaching aides. Contact numbers are
available on the attached author’s page.
Course
Overview
•••
The Senior High School
(SHS) curriculum
includes two core
subjects in
Mathematics that
students in the
eleventh grade are
required to take
regardless of their
chosen SHS career
track and strand.
These two core
subjects are General
Mathematics and
Statistics and
Probability with the
former being included
in the first term of the
school year followed
by the latter in the
second term.
The General
Mathematics course is
outlined with fifty-one
(51) competencies in
distributed over a
period of eighty (80)
hours during the whole
term. It has no
prerequisite course
and is divided into
three central Math
areas: (a) Functions
and their graphs; (b)
Basic business
mathematics; and (c)
Logic.
Introduction  1
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Learning Principles
in the K to 12 Basic Education Program
[excerpt from K to 12 Curriculum Guide in Mathematics (2013)]
The framework is supported by the following underlying learning principles and theories: Experiential and
Situated Learning, Reflective Learning, Constructivism, Cooperative Learning and Discovery and Inquirybased Learning. The mathematics curriculum is grounded in these theories.
Experiential Learning as advocated by David Kolb is learning that occurs by making sense of direct
everyday experiences. Experiential Learning theory defines learning as "the process whereby
knowledge is created through the transformation of experience. Knowledge results from the
combination of grasping and transforming experience" (Kolb, 1984, p. 41). Situated Learning, theorized
by Lave and Wenger, is learning in the same context in which concepts and theories are applied.
Reflective Learning refers to learning that is facilitated by reflective thinking. It is not enough that
learners encounter real-life situations. Deeper learning occurs when learners are able to think about
their experiences and process these, allowing them the opportunity to make sense of and derive
meaning from their experiences.
Constructivism is the theory that argues that knowledge is constructed when the learner is able to draw
ideas from his/her own experiences and connect them to new ideas.
Cooperative Learning puts premium on active learning achieved by working with fellow learners as they
all engage in a shared task.
The mathematics curriculum allows for students to learn by asking relevant questions and discovering
new ideas. Discovery Learning and Inquiry-based Learning (Bruner, 1961) support the idea that students
learn when they make use of personal experiences to discover facts, relationships, and concepts.
A complete copy of the curriculum guide for General mathematics may be found as part of the
appendices.
Learning Principles  2
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Functions
Week One, Functions and Their Graphs
Content Standards: The learner demonstrates understanding of key
concepts of functions.
Performance Standards: The learner is able to accurately construct
mathematical models to represent real-life situations using functions.
Learning Competencies: Over the course of one week, the learner is
expected to (1) represent real-life situations using functions,
including piece-wise functions [M11GM-Ia-1]; (2) evaluate a
function [M11GM-Ia-2]; (3) perform addition, subtraction,
multiplication, division, and composition of functions [M11GM-Ia-3];
and (4) solve problems involving functions [M11GM-Ia-4]
Learning Materials: In order to develop the targeted competencies,
the following materials are needed: (a) chalkboard and chalk; (b)
LCD projector and laptop or in the absence, visual materials.
Expected Outputs: In order to assess the attainment of the learning
competencies targeted, students will be required to undertake two
group performance tasks and two written works.
Procedure (Teacher’s Activity)
INTRODUCING FUNCTIONS
Day One [Target M11GM-Ia-1]
Represent real-life situations using functions, including piece-wise
functions.
Other Specific Objectives: The learner is also expected to attain at
least 75 percent proficiency in the following objectives: (a) explain
the concept of functions in comparison to relations; and (b)
compare the different types of functions learned in the previous
grades
1. Do routines and other preparatory activities for five minutes.
2. Present the lesson and the targeted competency (and other
learning outcomes).
Essential Ideas
How does it function?
•••
A function can be thought
of as a correspondence
from a set X of real
numbers x to a set Y of real
numbers y, where the
number y is unique for a
specific value of x.
It is the set of ordered pairs
of numbers (x, y) in which
no two distinct ordered
pairs have the same first
number. The set of all
admissible values of x is
called the domain of the
function, and the set of all
resulting values of y is
called the range of the
function (Leithold, 1996).
A piecewise-defined
function is one that is
defined by more than one
expression. These
expressions or pieces are
determined by restrictions
in the domain. This
function is also called a
split function because of
the behavior of its graph.
3. Facilitate activation of prior knowledge and motivation at the
same time. Accomplish this by doing the initial activity
described below.
Exploration Activity:
Will the relationship function?
The teacher may choose to relay the story in different ways but the use of visual materials is highly
suggested. Characters and even the actual situation may be altered to suit the interest of the
learners.
In this case, the story is relayed through a slideshow presentation
(https://prezi.com/q3a5mylivqfc/kasi-nga-senior-na-sila/).
At the end of the presentation, a question is posted for students to discuss. This will lead the class to
enrich their understanding of relations and functions.
At this point, it is necessary that the teacher directs the discussion so that the class is able to
process the following ideas: (1) a relation is any set of ordered pairs; (2) not all relations are
functions; and (3) only those whose domain do not include an abscissa that is shared by two or
more ordinates are considered functions.
Functions  3
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Once upon a time there were seven senior boys: Peter, James,
John, Matthew, Paul, Andrew and Mark. Close as they were, they
did almost everything together. So when they learned about the
school holding a Valentine’s Ball, they decided to find themselves
some date.
Peter
Janna
James
Valerie
John
Marcia
Incidentally, there were also seven beautiful senior girls who
were in the same class the boys were: Janna, Valerie, Marcia, April,
May June, and Julie. Our boys each had a crush on the girls and
they have decided to ask them out for dates.
Of the boys, Peter, James and John have something in
common. They always believe that honesty and trust make any
relationship strong. Peter asked Janna out for the date, James
chose Valerie and Marcia was asked by John.
April
Mark
May
June
Mark on the other hand thinks he needs to be on top of the
pack. He likes to collect then select. To make sure he gets first
date, he asked April, May and June without the three girls
knowing that all of them are being asked out by the same guy.
Matthew
Matthew, Paul and Andrew are very competitive and they all
liked Julie so they decided to ask her out and agreed that whoever
she picks, the other two would accept defeat.
Andrew
Paul
Julie
There are three kinds of relationships described in our story
but would all of these function?
4. Proceed by asking students the following questions.
a. What type of functions do you remember from your previous Math subjects?
b. Can you describe these functions?
5. Initiate discourse regarding equations being used as representations of functions. As initial
example, present the situation stated below.
The flag down rate
for metered taxis
in Metro Manila is
now down to PhP
30.00. For every
minute, the fare
goes up by PhP
5.00.
What are the two related variables mentioned in the situation presented?
The two related variables are time of travel and taxi fare.
Which variable is dependent and which is independent?
Taxi fare is dependent upon the time of travel which is the independent variable.
How much will a passenger pay if he rode the taxi for 20 minutes?
𝑃ℎ𝑃 5.00
(20 𝑚𝑖𝑛) (
) + 𝑃ℎ𝑃 30.00 = 𝑃ℎ𝑃 130.00
𝑚𝑖𝑛
What equation will best represent fare (F(t)) as a function of time (t) in minutes?
𝐹(𝑡) = 5𝑡 + 30
6. Utilize the other examples below to demonstrate the process. The same questions should be
asked in order to acquaint the students with the systematic way of representing functions
through equations.
For every box of cookies,
she sells in a month,
Anna donates a peso to
the Bantay Bata
Foundation. This and
the PhP 45.00 she saves
every month are put
into the foundation’s
bank account.
What are the two related variables mentioned in the situation presented?
The two related variables are number of boxes of cookies sold and donation.
Which variable is dependent and which is independent?
Anna’s donation depends on the number of boxes of cookies she sold in a month.
How big of a donation will the foundation receive from her this month if she
was able to sell 243 boxes of cookies this month?
(243)(𝑃ℎ𝑃 1.00) + 𝑃ℎ𝑃 45.00 = 𝑃ℎ𝑃 288.00
What equation will best represent the relationship between the two variables
described?
𝐷(𝑐) = 𝑐 + 45
Functions  4
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Mang Juan, who owns
the biggest meat shop
in the market has
offered a sale on pork. A
kilo costs PhP 160.00
but if you buy more than
3 kilos, each kilo is
priced at PhP 155.00.
What are the two related variables mentioned in the situation presented?
The price of the meat is related to the number of kilos bought.
Which variable is dependent and which is independent?
The number of kilos determines the price of the pork sold.
How much will you pay for five kilos of pork?
(5)(𝑃ℎ𝑃 155.00) = 𝑃ℎ𝑃 775.00
What equation will best represent the relationship between the two variables
described?
𝑃(𝑘) = {
160𝑘, 𝑘 ≤ 3
155𝑘, 𝑘 > 3
By this point, the teacher should describe piecewise functions as compared to other functions. Also
known as split functions, these functions contain multiple expressions called pieces that are used
depending on restrictions.
The next situation provides another example.
For Valentine’s day, SSG
officers launched a
fund-raising program
called “Dinner for a
Cause”. Each ticket is
worth PhP 120.00. If one
buys five tickets, he only
needs to pay PhP
550.00. If he buys more
than five tickets, the
price per ticket goes
down to PhP 105.00
What are the two related variables mentioned in the situation presented?
The amount and the number of tickets bought.
Which variable is dependent and which is independent?
The number of tickets bought determines how much one needs to pay.
How much will I save if I buy 7 tickets in one than just buying 5 tickets then
another 2?
[(2)(𝑃ℎ𝑃 120.00) + 𝑃ℎ𝑃 550.00] − [(7)(𝑃ℎ𝑃 105.00)] = 𝑃ℎ𝑃 55.00
What equation will best represent the relationship between the two variables
described?
120𝑡, 𝑡 < 5
𝑃(𝑡) = { 550, 𝑡 = 5
105𝑡, 𝑡 > 5
The following questions should be raised to aid generalization of the concept targeted: (a)
Describe a function in mathematics; (b) How do we represent functions through equations; (c)
What is a piecewise function? How is it different from the other types of function you have
previously encountered?
Similar situations and problems may be taken from The Calculus 7 by Louis Leithold (7th Edition).
Use some of these to further test students’ understanding.
PERFORMANCE TASK NO. 1
Evaluation/Assignment (Performance Task)
Functioning through Pieces
Divide the class into groups of four. Each group shall be asked to determine one situation that
produces a piecewise function. Through a written report, the group should be able to narrate
the situation, identify the variables, and finally, represent it through a function.
The rubric below should guide the teacher in evaluating the output of the students in this
performance task
Criteria
Quality of the
situation presented
Predictors
Doing okay
Can do better
The situation is neither
novel nor related to the
immediate environment
of the student. (3 credit)
Meeting expectations
The situation presented is
either novel or related to
the immediate
environment of the
student. (4 points)
The situation reflects both
novelty and real-life
environment. (5 points)
Functions  5
Von Christopher G. Chua
PERFORMANCE TASK NO. 1
General Mathematics Teaching Guides
•••
Clarity and
coherence of work
The situation lacks clarity
and is deemed confusing.
(2 point)
The situation is
understandable but may
be improved through
paraphrasing. (3 points)
Situation was narrated
clearly and is coherent
with the objective of the
task. (4 points)
Correctness of the
function presented
Function is entirely
incorrect based on the
narrated problem. (2
point)
Function may be further
improved through
simplification. (4 points)
Function is correct. (5
points)
Group dynamics
Members worked
independently or more
than one did not help with
the task at hand. (2 point)
One member clearly did
not participate. (3 points)
The project reflects
collaboration among
members of the group. (4
points)
Promptness in
Submission
Task completed past the
deadline. (no credit)
Score Interpretation
7 points or less
Task completed on or
before the set deadline (2
points)
8 to 14 points
15 to 20 points
Highest Possible Score: 20 points (2 per item)
Passing Score: 15 points (75 percent)
EVALUATING FUNCTIONS
Day Two [Target M11GM-Ia-2]
Evaluate a function
1. Do routines and other preparatory activities for five minutes.
2. Present the lesson and the targeted competency.
3. Facilitate activation of prior knowledge by using the following questions for recapitulation.
a. What is a function? How is it different from a mathematical relation?
b. What are peicewise-defined functions?
c. How are functions usually represented?
4. Present the problem stated here:
Mark has an internet shop as business. He charges PhP 20.00 for every hour that a customer uses a
computer. Being the wise businessman that he is, he always gets 10% of his daily income and saves it
to pay for electric charges. He also subtracts PhP 180.00 per day for his shop’s monthly rent. The rest
of the amount is the café’s income for the day.
a.
What are the two variables that may be determined from the problem? Which of these two is
the independent variable? [The computer shop’s income is dependent on the number of
hours of computer use per day]
b. Represent the café’s daily income through an equation.
[𝑓(𝑥) = 20𝑥 − (2𝑥 + 180) 𝑜𝑟 𝑓(𝑥) = 18𝑥 − 180]
c. How much would the café’s income be if it raked 80 hours’ worth of income? [PhP 1,300.00]
107 hours? [PhP 1,786.00]
d. What is least number of hours of computer use in the shop so that Mark gets his daily return
of investment? [10 hours]
Ask students to present their solutions for c and d on the board to be used for discussion.
5. Explain the process of evaluating functions based on the students responses to the problem stated
above. To better illustrate this process, use the concept of the function machine.
INPUT
(independent
variable, x)
FUNCTION
MACHINE
𝒇(𝒙)
OUTPUT
(dependent
variable, Y)
Functions  6
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
6. Discuss the next couple of examples to the class.
Let f be the function with
domain all real numbers x and
defined by the formula,
𝒇(𝒙) = 𝟑𝒙𝟑 − 𝟒𝒙𝟐 − 𝟑𝒙 + 𝟕.
Find f(2) iand f(-2).
𝑓(𝑥) = 3𝑥 3 − 4𝑥 2 − 3𝑥 + 7
𝑓(2) = 3(2)3 − 4(2)2 − 3(2) + 7
𝑓(2) = 3(8) − 4(4) − 3(2) + 7
𝑓(2) = 24 − 16 − 6 + 7
𝒇(𝟐) = 𝟗
If x represents the
temperature of an object in
degrees Celsius, then the
temperature in degrees
Fahrenheit is a function of x,
𝟗
given by 𝒇(𝒙) = 𝒙 + 𝟑𝟐.
Water freezes at 0°C and boils at
100°C. What are the corresponding
temperatures in °F?
9
𝑓(0) = (0) + 32 = 𝟑𝟐℉
5
9
𝑓(100) = (100) + 32 = 𝟐𝟏𝟐℉
5
𝟓
𝑓(𝑥) = 3𝑥 3 − 4𝑥 2 − 3𝑥 + 7
𝑓(−2) = 3(−2)3 − 4(−2)2 − 3(−2) + 7
𝑓(−2) = 3(−8) − 4(4) − 3(−2) + 7
𝑓(−2) = −24 − 16 + 6 + 7
𝒇(−𝟐) = −𝟐𝟕
Aluminum melts at 660°. What is its
melting point in °F?
9
𝑓(660) = (660) + 32 = 𝟏𝟐𝟐𝟎℉
5
For the more advanced students the next problem may be given as way of enrichment:
Given 𝑓(𝑥) = 2𝑥 2 + 3𝑥 − 9, evaluate 𝑓(𝑥 − 3)
𝑓(𝑥) = 2(𝑥 − 3)2 + 3(𝑥 − 3) − 9
𝑓(𝑥) = 2(𝑥 2 − 6𝑥 + 9) + 3(𝑥 − 3) − 9
𝑓(𝑥) = 2𝑥 2 − 12𝑥 + 18 + 3𝑥 − 9 − 9
𝒇(𝒙) = 𝟐𝒙𝟐 − 𝟗𝒙
7. The following question should be raised to aid generalization of the concept targeted: How do we
explain the process of evaluating a function?
8. Evaluate the function given at the values of the independent variable stated.
a. 𝑓(𝑥) = 2𝑥 2 − 7𝑥 + 9; 𝑓𝑖𝑛𝑑 𝑓(3) 𝑎𝑛𝑑 𝑓(−5) [f(3)=6; f(-5)=94]
b. 𝑔(𝑥) = 𝑥 − 8⁄2𝑥 + 1 ; 𝑓𝑖𝑛𝑑 𝑔(10) [2/21]
c. ℎ(𝑥) = 10𝑥 5 − 𝑥 4 + 3𝑥 3 − 7𝑥 2 − 10𝑥 + 1; 𝑓𝑖𝑛𝑑 ℎ(0)𝑎𝑛𝑑 ℎ(−1) [h(0)=1; h(-1)=-10]
WRITTEN WORK NO. 1
Evaluation (Pen and Paper Test)
Evaluate the following functions at the given value of the independent variable.
1. 𝑓(𝑥) = 8𝑥 − 11; 𝑓𝑖𝑛𝑑 𝑓(−5) [29]
2. 𝑓(𝑥) = 7𝑥 3 − 𝑥 + 3; 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑓(1) [9]
3.
𝑣
3
𝑚(𝑣) = 𝑣2 −3𝑣+2 ; 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑚(3) [2]
𝑟 + 7, 𝑟 ≤ 5
4. 𝑔(𝑟) = {
; 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑟(5)? [12]
19 − 𝑟, 𝑟 > 5
𝑟, 𝑟 < −3
5. 𝑔(𝑟) = { 𝑟 2 , 𝑟 = −3 𝑟(−2)? [3]
1 − 𝑟, 𝑟 > −3
Highest Possible Score: 15 points
Passing Score: 11 points (75 percent)
Scoring Guide:
Each item is good for three (3) points.
Give no point for no answer.
One point should be given for correct substitution only.
Three points is credited to correct answers.
ALGEBRA OF FUNCTIONS
Day Three [Target M11GM-Ia-3]
Perform addition, subtraction, multiplication, division, and composition of functions.
1. Do routines and other preparatory activities for five minutes.
2. Present the lesson and the targeted competency.
3. Facilitate activation of prior knowledge by using the following questions for recapitulation.
a. From our what we have discussed, what do you already know about functions?
b. How does the evaluation of a function work?
4. Divide the class into four sets. These sets will not necessarily work with each other but each set will
be given different tasks.
General Instruction: Given the expressions, 𝑥 2 − 8𝑥 − 20 and 𝑥 − 10, perform the operation assigned to
your set.
Functions  7
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Set A: Addition
[(𝑥 2 − 8𝑥 − 20) + (𝑥 − 10) = 𝒙𝟐 − 𝟕𝒙 − 𝟑𝟎]
Set B: Subtraction
[(𝑥 2 − 8𝑥 − 20) − (𝑥 − 10) = 𝒙𝟐 − 𝟗𝒙 − 𝟏𝟎]
Set C: Multiplication
[(𝑥 2 − 8𝑥 − 20)(𝑥 − 10) = 𝑥 3 − 8𝑥 2 − 20𝑥 − 10𝑥 2 + 80𝑥 + 200
= 𝒙𝟑 − 𝟐𝟖𝒙𝟐 + 𝟔𝟎𝒙 + 𝟐𝟎𝟎]
Set D: Division
𝑥 2 − 8𝑥 − 20
(𝑥 − 10)(𝑥 + 2)
[(
)=
= 𝒙 + 𝟐]
𝑥 − 10
(𝑥 − 10)
5. Ask one student from every set to discuss his/her answer to
class. Let the other students judge the solution and open up
the solutions for discussion.
𝑓(𝑥)
ቐ
‫۔‬
ۖ
‫ە‬
‫ۓ‬
ۖ
6. In order to introduce the notation for the operations on
functions, represent the given expressions as functions, that is,
Given the expressions, 𝑥 2 − 8𝑥 − 20 and 𝑥 − 10
𝑔(𝑥)
From the solutions of the students that are written on the
board, use the function notation to define each operation as
presented under “Essential Ideas”
7. Proceed with composition of functions. Have the students look
at the following functions,
𝑓(𝑚) = 𝑚2 − 3𝑚
𝑔(𝑚) = 𝑚 − 5
𝑔(𝑓(𝑚)) = 𝑚2 − 3𝑚 − 5
Essential Ideas
Operating Functions
•••
Definition of the Sum,
Difference, Product, and
Quotient of Two
Functions:
Given two functions, f and
g,
(i) their sum denoted by f +
g, is the function defined
by
(f + g)(x) = f(x) + g(x);
(ii) their difference,
denoted by f – g, is the
function defined by
(f - g)(x) = f(x) - g(x);
Ask the following questions:
What do you notice about the three functions? Can you see
any relationship existing among the functions given?
[Expected response: The value of n in g(n) when replaced by
f(m) will give us the third function.
Evaluating g(n) at f(m) will result to g(f(m).]
Explain that the third function is called a composite function of
the first two functions. Show the process of obtaining the third
function from the first two using the correct notation for
composite functions.
(iii) their product denoted
by f · g, is the function
defined by
(f · g)(x) = f(x) · g(x);
(iv) their quotient denoted
by f/g, is the function
defined by
(f / g)(x) = f(x) / g(x)
Definition of a Composite
Function
(𝑔 ∘ 𝑓)(𝑚) = 𝑔(𝑓(𝑚))
𝑔(𝑓(𝑚)) = (𝑚2 − 3𝑚) − 5
(𝑔 ∘ 𝑓)(𝑚) = 𝑚2 − 3𝑚 − 5
8. The following question should be raised to aid generalization of
the concept targeted: What are the five operations on
functions? State each in the general function notation.
9. Work with the following functions:
𝑓(𝑥) = 𝑥 3 − 1
𝑔(𝑥) = 𝑥 2 + 2𝑥 + 1
ℎ(𝑥) = 𝑥 + 1
a. (𝑓 + 𝑔 + ℎ)(𝑥) [𝒙𝟑 + 𝒙𝟐 + 𝟑𝒙 + 𝟏]
b. (𝑓 − ℎ)(𝑥) [𝒙𝟑 − 𝒙 − 𝟐]
𝑓⋅𝑔
c. (
) (𝑥) [(𝑥 3 − 1)(𝑥 + 1) 𝑜𝑟 𝒙𝟒 + 𝒙𝟑 − 𝒙 − 𝟏]
ℎ
d. (𝑔 ∘ ℎ)(𝑥) [𝑥 2 + 2𝑥 + 1 + 2𝑥 + 2 + 1 = 𝒙𝟐 + 𝟒𝒙 + 𝟓]
For the more advanced students the next problem may be
given as way of enrichment:
With the same functions what is (ℎ ∘ 𝑓)(3𝑥)?
[(ℎ ∘ 𝑓)(𝑥) = 𝑥 3 ; 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 (ℎ ∘ 𝑓)(3𝑥) = 𝟐𝟕𝒙𝟑 ]
Given the two functions f
and g the composite
function, denoted by 𝒇 ∘ 𝒈
is defined by
(𝒇 ∘ 𝒈)(𝒙) = 𝒇(𝒈(𝒙)
And the domain of 𝒇 ∘ 𝒈 is
the set of all numbers x in
the domain of g such that
g(x) is on the domain of f.
(Leithold, 1996)
Functions  8
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Assignment (Pen and Paper Test)
WRITTEN WORK NO. 2
Three functions are defined as follows: 𝑚(𝑥)
= −2𝑥 + 7; 𝑛(𝑥) = 𝑥 4 + 𝑥 2 − 2; 𝑝(𝑥) =
𝑥+1
.
𝑥−3
Define/derive the following functions:
Scoring Guide:
1. (𝑚 + 𝑛)(𝑥) [𝑥 4 + 𝑥 2 − 2𝑥 + 5]
Each item is good for three (3) points.
2. (𝑛 − 𝑚)(𝑥) [𝑥 4 + 𝑥 2 + 2𝑥 − 9]
2
Give no point for no answer.
−2𝑥 +5𝑥+7
3. (𝑚 ⋅ 𝑝)(𝑥) [
]
One point should be given for correct function notation.
𝑥−3
Two for correct substitution of the functions involved.
𝑥+1
𝑝
4. ( ) (𝑥) [
]
2
Three points is credited to correct answers.
𝑚
−2𝑥 +13𝑥−21
5. (𝑚 ∘ 𝑛)(𝑥) [−2𝑥 4 − 2𝑥 2 + 11]
6. (𝑚 − 𝑛)(−2) [−(−2)4 − (−2)2 − 2(−2) + 9 = −𝟕]
Highest Possible Score: 18 points
Passing Score: 14 points (78 percent)
FUNCTIONS AS MATHEMATICAL MODELS
Day Four [Target M11GM-Ia-4]
Solve problems involving functions
1. Do routines and other preparatory activities for five minutes.
Collect the assignment and discuss items that need to be
discussed.
2. Present the lesson and the targeted competency.
3. Facilitate activation of prior knowledge by using the following
questions for recapitulation.
a. From our what we have discussed, what do you already
know about functions?
b. What are the different operations on functions? Describe
each.
4. Discuss the five suggestions for solving problems involving
functions as mathematical models according to Leithold
(1996).
5. Demonstrate the process of solving the following problems
using functions as mathematical models.
PROBLEM NO. 1
A wholesaler sells a product by the kilo (or fraction of a kilo). If
not more than 10 kilograms are ordered, the wholesaler charges
PhP 200.00 per kilo. However, to invite large orders, the
wholesaler charges only PhP 180.00 per pound if more than 10
kilograms are ordered.
Find a mathematical model expressing the cost of the order as a
function of the amount of the product ordered.
In the given situation, the cost of an order is dependent on the
number of kilos of the product.
Represent these two variables by letting C(x) be the cost of
ordering x kilos of the product.
Since the cost is calculated differently when not more than 10
kilos is ordered and when the order exceeds 10 kilos, the
function is a piecewise-defined function.
For orders not exceeding 10 kilos, 𝐶(𝑥) = 200𝑥
For orders more than 10 kilos, 𝐶(𝑥) = 180𝑥
𝐶(𝑥) = {
200𝑥,
180𝑥,
𝑖𝑓 0 ≤ 𝑥 ≤ 10
𝑖𝑓 𝑥 > 10
Essential Ideas
Making Problem Solving
less of a problem
•••
Suggestions for Solving
Problems Involving a
Function as Mathematical
Model (Leithold, 1996)
1. Read the problem carefully
so that you understand it.
Make up a specific example
that involves a similar
situation in which all the
quantities are known.
Another aid is to draw a
picture.
2. Determine the known and
unknown quantities.
Represent the independent
variable and the function
that is obtained.
3. Write down any numerical fact
known about the variable and
the function value.
4. From the information in step
3, determine two algebraic
expressions for the same
number, one in terms of the
variable and one in terms of
the action value. From these
two expressions, form an
equation that defines the
function
5. After applying the
mathematical model to solve
for the unknown quantities,
write a conclusion that
answers the question of the
problem. Be sure this contains
the correct unit of measure.
Functions  9
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Determine the total cost of an order of 9.5 kilos and of an order
of 10.5 kilos.
For 𝑥 = 9.5 𝑘𝑖𝑙𝑜𝑠, 𝐶(9.5) = 200(9.5) = 1900
For 𝑥 = 10.5 𝑘𝑖𝑙𝑜𝑠, 𝐶(10.5) = 180(10.5) = 1890
PROBLEM NO. 2
A cardboard box manufacturer wishes to make open boxes from
rectangular pieces of cardboard with dimensions 10 in. by 17 in.
by cutting equal squares from four the four corners and turning
up the sides.
Find a mathematical model expressing the volume of the box as
a function of the length of the side of the square cut out.
Since the length of the side of the squares is relative, we
represent it with x and shall determine the volume of the
rectangular open boxes represented by V(x).
The volume of a rectangular prism is length x width x height.
Therefore, 𝑉(𝑥) = (10 − 2𝑥)(17 − 2𝑥)𝑥
Simplified, 𝑽(𝒙) = 𝟒𝒙𝟑 − 𝟓𝟒𝒙𝟐 + 𝟏𝟕𝟎𝒙
6. To serve both as application and performance task, assign the activity described below.
Evaluation/Assignment (Performance Task)
In groups of five, students will need to solve any two of the five problems stated below. Answers
should be written on a whole sheet of paper.
PERFORMANCE TASK NO. 2
1. An ice cream vendor makes a profit of 𝑃(𝑥) = 7𝑥 − 525 when selling 𝑥 scoops of ice cream
per day. How many scoops of ice cream should be sold for break-even sales (𝑃(𝑥) = 0)? How
much profit will the vendor earn for selling 235 scoops of ice cream?
2. A senior high school student earns income through encoding for which she charges PhP
10.00 a page. However, she gives a 5 percent discount if the encoding job exceeds 20
pages. Represent how much she charges per encoding job as a function of the number of
pages per job. How much would she earn for encoding 29 pages?
3. A cellular phone company estimates that if it has 𝑥 thousand subscribers, then its monthly
profit is 𝐸(𝑥) = 671𝑥 − 53,032. How many subscribers are needed for a monthly profit of PhP
766,259.00? How much will the company earn if it has 13,799,000 subscribers in one month?
4. The regular adult admission price to an evening performance at a cinema is PhP 300.00
while the price for children under 12 years of age is PhP 200.00 and the price for senior
citizens (60 or older) is PhP 225.00. Find a mathematical model expressing the price as a
function of the person’s age. How much will one pay for 7 tickets if two of these are for
children, one for a senior, and the rest are for regular adults?
5. The cost of a cellular phone call for a telecom is at PhP 6.00 for the first minute and PhP 4.50
for every minute after the first. How much would an eight-minute long call cost you? Express
the relationship between the two variables through a function.
Highest Possible Score: 20 points (10 per item)
Passing Score: 15 points (75 percent)
Find time to discuss the answers to the problems.
As a means of wrapping up the topics discussed over the week, use the following
questions for generalization: What are functions? How are functions different from relations? How are
they evaluated? When performing operations with functions, how would you describe each process?
Why is there a need to study functions? What is the advantage of knowing how to create mathematical
models through functions?
Functions  10
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Rational Functions
Week Two, Functions and Their Graphs
Content Standards: The learner demonstrates understanding of key
concepts of functions.
Performance Standards: The learner is able to accurately formulate
and solve real-life problems involving rational functions.
Learning Competencies: Over the course of one week, the learner is
expected to (1) represent real-life situations using rational functions
[M11GM-Ib-1]; (2) distinguish rational function, rational equation,
and rational inequality [M11GM-Ib-2]; (3) find the domain and
range of a rational function [M11GM-Ib-5]; (4) determine the: (a)
intercepts, (b) zeroes, and (c) asymptotes of rational functions
[M11GM-Ic-1]; (5) graph rational functions [M11GM-Ic-2]; and (6)
represent a rational function through its: (a) table of values (b)
graph, and (c) equation [M11GM-Ib-4]
Important Note: For an improved continuity of competencies under the same
content standard, the learning competencies have been rearranged but the
codes have been maintained for reference.
Learning Materials: In order to develop the targeted competencies,
the following materials are needed: (a) chalkboard and chalk; (b)
LCD projector and laptop or in the absence, visual materials; (c)
calculators
Expected Outputs: In order to assess the attainment of the learning
competencies targeted, students will be required to undertake two
group performance tasks and one written work.
Procedure (Teacher’s Activity)
RATIONAL FUNCTIONS
Day One [Target M11GM-Ib-1 and
M11GM-1b-2]
Represent real-life situations using rational functions and
distinguishes rational function, rational equation, and rational
inequality
Other Specific Objective: The learner is also expected attain at least
75 percent proficiency in comparing rational functions with other
types of functions.
Essential Ideas
Polynomial versus Rational
•••
An algebraic function is
one formed by a finite
number of algebraic
operations on the identity
function and a constant
function. These algebraic
operations include
addition, subtraction,
multiplication, division,
raising to powers, and
extracting roots.
If a function f is defined by
𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 +
⋯ + 𝑎1 𝑥 + 𝑎0
where 𝑎0 , 𝑎1 , … , 𝑎𝑛 are
real numbers and n is a
nonnegative integer, then f
is called a polynomial
function of degree n.
If a function can be
expressed as the quotient
of two polynomial
functions, it is called a
rational function.
(Leithold, 1996)
Said differently, r is a
rational function if it is of
𝑝(𝑥)
the form, 𝑟(𝑥) = 𝑞(𝑥),
where p and q are
polynomial functions.
1. Do routines and other preparatory activities for five minutes.
2. Present the lesson and the targeted competencies (and other
learning outcome).
3. Facilitate activation of prior knowledge with these questions:
a. What are functions?
b. What are the different types of functions? Differentiate them.
4. Present the following couple of sets of functions to class. Ask the students what they think was the
basis/rule for grouping the functions.
Exploration Activity:
Grouped how?
Group A
Group B
6
𝑓(𝑥) = 5𝑥 − 2
𝑓(𝑥) =
2
2𝑥 − 3
𝑔(𝑥) = 𝑥 + 3𝑥 + 8
Rational Functions  11
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
2 5
𝑥 + 8𝑥 3 − 𝑥 − 18
3
𝑥3 + 𝑥2 − 𝑥 − 1
𝑚(𝑥) =
10
𝑥−5
𝑥+2
12𝑥 + 8
ℎ(𝑥) = 2
𝑥 +𝑥+1
𝑥3 + 𝑥2 − 𝑥 − 1
𝑚(𝑥) =
𝑥−1
[Expected response: Functions in Group B are those that have variables as denominators while
those in Group A don’t.]
Teacher’s Question: One of these two groups of functions is composed of rational functions. Which of the two
groups do you think is it? Explain the basis for your answer.
ℎ(𝑥) =
𝑔(𝑥) =
5. Use students responses in the previous item to differentiate polynomial and rational functions. If
necessary present some other examples to deepen understanding.
6. The concept of functions, equations, and inequalities are not entirely new to the students at this
level. Use their prior knowledge and ask them to construct concept maps by groups of threes. You
may choose to implement a standard form of the concept map such as the one provided below.
Exploration Activity:
Know your Circles…
Sort out the keywords/ mathematical statements provided by writing them
inside the appropriate circles found in the diagram below.
(1)
PERFORMANCE TASK NO. 3
(3)
•
•
•
•
•
•
•
•
•
•
(2)
KEYWORDS:
Function
Equation
Inequality
Dependent and independent
variables
Equal sign
Greater or lesser
Shaded graphs
Graphs formed by lines and
curves
One to many
One to one
• 𝑦=
𝑥
𝑥−3
• 𝑥𝑦 − 7𝑦 = 9 − 2𝑥
2
• 𝑦≥
6𝑥−7
Key to Correction:
EQUATION
Equal Sign, Graphs formed
by lines and curves, one to
many, 𝑥𝑦 − 7𝑦 = 9 − 2𝑥
FUNCTION
Dep Ind var,
one to one,
𝑦=
INEQUALITY
Greater or lesser,
shaded graphs, one to
many, 𝑦 ≥
2
6𝑥−7
𝑥
𝑥−3
Supplement students’ responses in the activity by discussing their answers as a class. Have them
defend their answers and ask them provide other mathematical statements that conform to
each kind determined above.
7. Ask the following questions for wrap-up:
a. What are rational functions?
b. How would you determine if a functional is rational and not polynomial?
c. How would you differentiate rational equations, rational functions, and rational inequalities?
Rational Functions  12
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
8. Are the following functions rational?
a.
𝑓(𝑥) =
b.
𝑔(𝑏) =
c.
𝑚(𝑡) =
3𝑥 2 +5𝑥−7
[R]
𝑥
𝑏5 −𝑏4 +8𝑏3 −3𝑏2 +𝑏
𝑡−13
9
13
𝑏
[P, the function may be simlplified by division]
[P, no variable in the denominator]
d.
𝑘(𝑟) = + 5𝑟 − 7𝑟 2 [R]
e.
𝑓(𝑥) = (6𝑥 − 1)−2 [R]
𝑟
DOMAIN AND RANGE
Day Two [Target M11GM-Ib-5]
Find the domain and range of a rational function.
SOLVING RATIONAL FUNCTIONS
1. Do routines and other preparatory activities for five minutes.
2. Present the lesson and the targeted competency.
3. Facilitate activation of prior knowledge with these questions:
a. What are rational functions?
b. What is meant by the domain of a function? the range of a function?
4. The first half of the discussion shall be spent on inquiry – one where students will be asked a series
of questions and they have to answer as a class. Their answers to the questions will help them
come up with the general idea of how to determine restrictions in the allowable values of the
independent variable, 𝑥.
What follows would be a detailed simulation of the class discussion facilitated by the teacher.
Student Activity
Teacher Activity
(Answers should vary. Adjust accordingly)
I would like to ask somebody to provide an example
[Gives a rational function. In this case, let us assume
of a rational function, one which has a linear
𝑥+4
expression as its denominator. Any volunteer?
that the function is 𝑓(𝑥) =
]
𝑥−3
(Call one student.)
Let’s consider the function given by (name of
student). Can 𝑥 take any number for its value?
[Students provide suggestions]
(For every answer, ask the student why s/he thinks that 𝑥 cannot take 3 for its value because it will make the
the number should be a restriction for 𝑥. Do this until
denominator zero. If this happens, the value of 𝑦
the correct answer is suggested and properly
becomes undefined.
explained.)
So how would you describe the domain of the
function 𝑓(𝑥)?
The domain is the set of all real numbers except 3.
(Call another student.)
Correct.
{𝑥|𝑥 ∈ ℝ, 𝑥 ≠ 3}
Can anybody state this in set notation?
(Call another student.)
[Gives the second rational function. In this case, let us
Let’s discuss another function. Who can give me
another rational function but this time, with a
quadratic expression as denominator?
(Call another student.)
Good. We shall do the same with this function. What
values of 𝑥 will make the function undefined?
How do we express the domain of the function in set
builder notation?
assume that the function is 𝑓(𝑥) =
5
]
𝑥 2 +4𝑥+3
It is preferable to consider examples with
denominators that are factorable. If it happens that
the student comes up with an expression with
irrational roots, use the appropriate method to
identify its zeroes.
We cannot take the values of -3 and -1 for 𝑥.
{𝑥|𝑥 ∈ ℝ, 𝑥 ≠ −3, −1}
5. Next, we discuss the how to define the range of a rational function. For this process, there is a
need for students to be taught how to manipulate the function in order to find the restrictions.
Important Note: The teacher should plan the functions to be discussed very carefully as many functions would
require higher mathematical skills. Restrict examples depending on the capacity and readiness of the students.
Use the examples previously provided by the students. The complete solutions are shown below.
Rational Functions  13
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Emphasize here that the main objective in order to define the range is to express the function
explicitly in 𝑥 in terms of 𝑦.
𝒇(𝒙) =
𝑦=
𝑥 − 3 (𝑦 =
𝑥+4
𝑥−3
𝑥+4
)𝑥 − 3
𝑥−3
𝒙+𝟒
𝒙−𝟑
𝒇(𝒙) =
Change the function
notation into the
dependent variable y.
Multiply both sides by
the denominator of
the right side.
𝑦=
5
2
𝑥 + 4𝑥 + 3
𝑥 2 + 4𝑥 + 3 =
𝑥 2 + 4𝑥 + 4 =
Simplify
𝑥𝑦 − 𝑥 = 3𝑦 + 4
Isolate all terms with 𝑥
on the left side of the
equation and those
without it to the right
side.
𝑥 2 + 4𝑥 + 4 =
𝑥(𝑦 − 1) = 3𝑦 + 4
Factor out 𝑥 on the left
side.
(𝑥 + 2)2 =
Divide both sides of
the equation by the
other factor.
3𝑦 + 4
𝑦−1
{𝒚|𝒚 ∈ ℝ, 𝒚 ≠ 𝟏}
Finally, determine the
restriction through set
builder notation.
𝑥=
5
𝑦
5
+4−3
𝑦
𝑥𝑦 − 3𝑦 = 𝑥 + 4
𝑥=
𝒙𝟐
5
+1
𝑦
𝟓
+ 𝟒𝒙 + 𝟑
Change the function
notation into the
dependent variable y.
Multiply both sides by
the denominator and
divide both sides by 𝑦
Transpose the
constant, complete
the square and add
the third term of the
perfect square
trinomial to the right
side of the equation.
Simplify.
Express the perfect
square trinomial as a
square of a binomial.
Extract the roots,
rationalize, and
transpose the
constant.
5
+1
𝑦
√5𝑦 + 𝑦 2
−2
𝑦
{𝒚|𝒚 ∈ ℝ, −𝟓 ≥ 𝒚 > 𝟎}
Finally, determine the
restriction through set
builder notation.
Ask the following questions for wrap-up.
a. What is meant by the domain of a function? The range of a function?
b. How do we define the domain of a rational function?
c. Describe the process of determining the restrictions for the range of a rational function.
Have the students perform a drill on the targeted competency. Let them find the domain of the
following rational functions.
2𝑥−1
a.
𝑓(𝑥) =
b.
𝑔(𝑥) = 2 − 𝑥+1 [𝑥 ≠ −1]
c.
ℎ(𝑥) =
d.
𝑥+1
[𝑥 ≠ −1]
2𝑥 2 −1
𝑟(𝑥) =
𝑥 2 −1
3
3𝑥−2
− 𝑥 2 −1 [𝑥 ≠ ±1]
2𝑥2 −1
𝑥2 −1
3𝑥−2
𝑥2 −1
2
[𝑥 ≠ ±1, ]
3
INTERCEPTS, ZEROS, ASYMPTOTES
Day Three [Target M11GM-Ic-1]
Determine the: (a) intercepts, (b) zeroes, and (c) asymptotes of rational SOLVING
functions.
RATIONAL FUNCTIONS
1. Do routines and other preparatory activities for five minutes.
2. Present the lesson and the targeted competency.
3. Facilitate activation of prior knowledge with these questions:
a. How would you describe the domain and range of a rational function?
b. What do you think happens to the graph of the rational function at the values for which it
becomes undefined?
The second question should allow students to make assumptions in relation to the competency
targeted. Take note and emphasize their hypotheses and have them look after the discussion.
Rational Functions  14
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
4. Project (or post on the
board) the graph of
𝑥+2
𝑓(𝑥) =
as shown at
𝑥−2
the right.
Point out the
intercept, zero, and
the asymptotes of the
graph of the function.
Define each of these
terms as they relate to
the graph.
Essential Ideas
Intercepts, Zeros,
Asymptotes
horizontal
asymptote
zero
•••
intercept
5. Ask the following
questions:
a. It is obvious that
the zero of the
𝒙+𝟐
function is a point
Graph of 𝒇(𝒙) =
vertical
𝒙−𝟐
asymptote
on the x-axis so
the y-coordinate is zero. How do we look for its xcoordinate? [substitute zero to the value of 𝑦 in the function then
solve for the value of 𝑥.]
𝑥+2
0=
; 𝑥 + 2 = 0; 𝒙 = −𝟐
𝑥−2
b. How do we determine the intercept? [substitute zero to the
value of x in the function then solve for the value of 𝑦.]
0+2
2
𝑦=
;𝑦 =
; 𝒚 = −𝟏
0−2
−2
c. How do we determine the vertical asymptote based on
the equation? The vertical asymptote of the graph is at 𝑥 =
2. How is this related to the domain of the function? [The
function is undefined for 𝑥 = 2. It is a restriction for the domain.]
A point at which the graph
crosses the y-axis is called
a y-intercept, and a point at
which it crosses the x-axis
is called an x-intercept.
The x-coordinate of an xintercept is sometimes
called a zero of the
function since the function
has a zero value there.
The line 𝑥 = 𝑐 is called a
vertical asymptote of the
graph of a function 𝑦 =
𝑓(𝑥) if as 𝑥 → 𝑐 − or as
𝑥 → 𝑐 + , either 𝑓(𝑥) →
+∞ or 𝑓(𝑥) → −∞.
The line 𝑦 = 𝑐 is called a
horizontal asymptote of the
graph of a function 𝑦 =
𝑓(𝑥) if as 𝑥 → −∞ or as
𝑥 → ∞ , either 𝑓(𝑥) → 𝑐.
Let us find out what happens to the function as our xcoordinate gets closer and closer to 2. Complete the table
below.
𝒙
𝒇(𝒙)
𝒙
𝒇(𝒙)
1
-3
3
5
1.5
-7
2.5
9
1.9
-39
2.1
41
1.99
-399
2.01
401
1.999
-3,999
2.001
4,001
1.9999
-39,999
2.0001
40,001
What happens to 𝑓(𝑥) as we get closer and closer to 2 from the left? [The function gets lesser and
lesser values.] From the right? [The function increases.]
Teacher: This is the reason why the graph has an asymptote at that portion. As we get closer to the value of
2, we get larger and larger values or smaller and smaller values but we will never get an actual value of the
function when 𝑥 = 2.
d. How do we determine the horizontal asymptote of the graph of the function? Do you think this
may be related to the restriction in our range? [The function has a horizontal asymptote at 𝑦 = 1
since there is no value for 𝑥 that would give us a function value of 1.]
𝑥+2
𝑥+2
𝟐𝒚 + 𝟐
𝑓(𝑥) =
; 𝑦=
; 𝑥𝑦 − 2𝑦 = 𝑥 + 2; 𝑥𝑦 − 𝑥 = 2𝑦 + 2; 𝑥(𝑦 − 1) = 2𝑦 + 2; 𝒙 =
; 𝒚≠𝟏
𝑥−2
𝑥−2
𝒚−𝟏
Use the following statements for wrap-up:
a. Define the following: intercept, zero, asymptote.
b. How do we find the intercept of a rational function?
c. How do we determine the zero(s) of a rational function?
d. What process can we do to identify the location of the asymptotes of a rational function?
Rational Functions  15
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Allow students to work with a similar task
together with a partner. Give the function
2𝑥
𝑔(𝑥) =
for which they have to identify
𝑥+3
the intercept(s), the zero(s), and the
asymptotes.
[The intercept and zero are the same. The graph
passes through the origin. The vertical asymptote
is at 𝑥 = −3 while the horizontal asymptote is 𝑦 =
2]
Show the graph as a means to confirm the
answers of the students. Discuss in class.
WRITTEN WORK NO. 3
Assignment (Pen and Paper Test)
Identify the intercept(s), the zero(s), and
the asymptotes of the rational functions.
Graph of 𝒇(𝒙)
=
𝟐𝒙
𝒙+𝟑
1
1. 𝑓(𝑥) = [𝑛𝑜𝑛𝑒, 𝑛𝑜𝑛𝑒, 𝑥 − 𝑎𝑥𝑖𝑠, 𝑦 − 𝑎𝑥𝑖𝑠]
𝑥
2. 𝑔(𝑥) =
3. 𝑚(𝑥) =
3𝑥
𝑥−5
Scoring Guide:
For every item, give one point each for the correct
intercept and the correct zero, two points for each
correct asymptote.
Solutions are necessary.
[0, 0, 𝑥 = 5, 𝑦 = 3]
2𝑥−7
7𝑥−1
1
2
7
7
[7, 7, 𝑥 = , 𝑦 = ]
Highest Possible Score: 18 points
Passing Score: 14 points (78 percent)
GRAPHS OF RATIONAL FUNCTIONS
Day Four [Target M11GM-Ic-2 and M11GM-Ib-4]
Graph rational functions and represent a rational function through its: (a) table of values (b) graph, and
(c) equation.
1. Do routines and other preparatory activities for five minutes.
Collect the assignment and discuss items that need to be
discussed.
2. Present the lesson and the targeted competencies.
3. Facilitate activation of prior knowledge by using the question
for recapitulation.
a. What are the three significant features of the graphs of
rational functions that we discussed? Describe each of
those.
4. Discuss the five steps in graphing rational functions. Provide
𝑥+2
the function, 𝑓(𝑥) =
as initial example.
𝑥−2
If necessary, use another function, 𝑔(𝑥) =
4−𝑥
𝑥+1
as second
example. For every step, ask students to participate in order
to give them the chance to assess the steps on their own.
In constructing the graphs, start by locating the asymptotes,
the intercept, and the zero. Then use the table to determine
the behavior of the graph as it gets closer to the vertical
asymptote.
Essential Ideas
Graphing Rational
Functions
•••
To graph 𝑓(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
where p
and q have no common factor
other than 1,
1. determine x-intercepts by
solving 𝑓(𝑥) = 0.
2. determine the y-intercept by
evaluating 𝑓(0).
3. determine the vertical
asymptotes by solving 𝑞(𝑥) = 0
4. determine the horizontal
asymptote (if any) by dividing by
the highest power of x in the
denominator.
5. plot additional points choosing
at least one value of x from each
interval determined by the xintercepts or vertical asymptotes
Rational Functions  16
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Example, 𝒇(𝒙) =
Procedure
STEP ONE. Determine xintercepts by solving 𝑓(𝑥) = 0.
STEP TWO. Determine the yintercept by evaluating 𝑓(0).
STEP THREE. Determine the
vertical asymptotes by solving
𝑞(𝑥) = 0
STEP FOUR. Determine the
horizontal asymptote (if any) by
dividing by the highest power of
x in the denominator.
𝒙+𝟏
𝒙−𝟑
𝑥+1
0=
; 𝑥 + 1 = 0; 𝒙 = −𝟏
𝑥−3
0+1
1
𝑓(0) =
=−
0−3
3
𝑥 − 3 = 0; 𝑥 = 3
𝑥 1
1
+
1+
𝑥
𝑥
𝑥; 𝑦 = 1
𝑓(𝑥) =
=
𝑥 3
3
−
1−
𝑥 𝑥
𝑥
𝑥
𝑓(𝑥)
𝑥<3
1.5
2
2.5
-1.25
-3
-7
Decreasing towards 3
𝑥>3
3.5
4
4.5
9
5
3.67
Increasing towards 3
STEP FIVE. Plot additional points
choosing at least one value of x
from each interval determined
by the x-intercepts or vertical
asymptotes
Example, 𝑔(𝑥) =
Procedure
STEP ONE. Determine xintercepts by solving 𝑔(𝑥) = 0.
STEP TWO. Determine the yintercept by evaluating 𝑔(0).
STEP THREE. Determine the
vertical asymptotes by solving
𝑞(𝑥) = 0
STEP FOUR. Determine the
horizontal asymptote (if any) by
dividing by the highest power of
x in the denominator.
4−𝑥
𝑥+1
4−𝑥
0=
; 4 − 𝑥 = 0; 𝒙 = 𝟒
𝑥+1
4−0
𝑔(0) =
=4
0+1
𝑥 + 1 = 0; 𝑥 = −1
4 𝑥 4
−
−1
𝑔(𝑥) = 𝑥 𝑥 = 𝑥
; 𝑦 = −1
𝑥 1
1
+
1+
𝑥 𝑥
𝑥
𝑥
𝑔(𝑥)
𝑥 < −1
-2.5
-2
-1.5
-4.33
-6
-11
Decreasing towards -1
𝑥 > −1
-0.5
0
0.5
9
4
2.33
Increasing towards -1
STEP FIVE. Plot additional points
choosing at least one value of x
from each interval determined
by the x-intercepts or vertical
asymptotes
Rational Functions  17
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
To serve both as application and performance task, assign the activity described below.
Evaluation/Assignment (Performance Task)
In groups of seven, students will need to graph the following rational functions by following the
steps previously discussed. Ask one representative from each group to pick a number that would
determine the function they will have to work with. This will ensure fairness in the assignment.
𝑥−2
GROUP A. 𝑓(𝑥)
= 2𝑥−3
GROUP B. 𝑓(𝑥)
2𝑥−1
=
GROUP E. 𝑔(𝑥)
GROUP F. 𝑔(𝑥)
𝑥+2
=
=
3𝑥
GROUP G. ℎ(𝑥)
−2
GROUP H. ℎ(𝑥)
GROUP C. 𝑓(𝑥)
= 𝑥−5
GROUP D. 𝑔(𝑥)
= 3−𝑥
3𝑥−1
𝑥
3−2𝑥
𝑥+1
5𝑥
= 𝑥+5
𝑥+2
= 2𝑥+3
PERFORMANCE TASK NO. 4
Use the rubric below as guide in evaluating students’ outputs.
Criteria
Can do better
Predictors
Doing okay
Meeting expectations
Correctness and
accuracy in
following graphing
procedure.
At least two of the steps
were not properly carried
out (3 points)
One of the steps in the
procedure was incorrectly
done (4 points)
All steps in the procedure
were followed and
properly employed (5
points)
Quality of Graph
The graph was not
accurately sketched with
two or more features not
characterized (3 point)
One important feature of
the graph is not correct (4
points)
The graph is correct (5
points)
Cleanliness and
completeness of
output.
Erasures are evident in the
work submitted (no
credit)
Output is free of erasures
and unnecessary writings
(2 points)
Group dynamics
Members worked
independently or more
than one did not help with
the task at hand. (2 point)
The project reflects
collaboration among
members of the group. (5
points)
Time Management
Task completed past the
intended time. (no credit)
Score Interpretation
8 points or less
One member clearly did
not participate. (3 points)
Task completed on or
before the intended time
(3 points)
9 to 16 points
17 to 20 points
Find time to discuss the graphs constructed by the students.
As a means of wrapping up the topics discussed over the week, use the following
questions for generalization:
a. What are rational functions? How are they different/ related to polynomial functions?
b. How do you determine if a function is rational in nature?
c. What is meant by the intercept of the graph of a rational function? How do we locate the
intercept of the graph of a rational function?
d. Describe asymptotes?
e. What is the significance of restrictions in the domain and range of a rational function?
f. How do we graph a rational function?
Rational Functions  18
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Rational Equations and Inequalities
Week Three, Functions and Their Graphs
Content Standards: The learner demonstrates understanding of key
concepts of functions.
Essential Ideas
Sign Diagrams
Performance Standards: The learner is able to accurately formulate
and solve real-life problems involving rational functions.
•••
Steps for constructing a
sign diagram for a rational
function:
Learning Competencies: Over the course of one week, the learner
is expected to (1) solve rational equations and inequalities
[M11GM-Ib-3]; and (2) solve problems involving rational equations
and inequalities [M11GM-Ic-3]
Suppose r is a rational
function.
Important Note: For an improved continuity of competencies under the same
content standard, the learning competencies have been rearranged but the
codes have been maintained for reference.
1. Place any value
excluded from the domain
of r on the number line
with an “!” above them.
Learning Materials: In order to develop the targeted competencies,
the following materials are needed: (a) chalkboard and chalk; (b)
LCD projector and laptop or in the absence, visual materials; (c)
calculators
2. Find the zeros of r and
place them on the number
line with the number “0”
above them.
Expected Outputs: In order to assess the attainment of the learning
competencies targeted, students will be required to undertake one
group performance task and one written work.
3. Choose a test value in
each of the intervals
determined in steps 1 and
2.
Procedure (Teacher’s Activity)
SOLVING RATIONAL EQUATIONS
Day One [Target M11GM-Ib-3]
Solve rational equations and inequalities
4. Determine the sign of
r(x) for each test value in
step 3, and write that sign
above the corresponding
interval.
1. Do routines and other preparatory activities for five minutes.
2. Present the lesson and the targeted competency.
3. Facilitate activation of prior knowledge with this question:
a. How would you compare functions, equations, and
inequalities?
4. Discuss in detail the solutions for solving the following rational
equations and inequalities.
Emphasize that in solving rational equations, the technique is to rid the equation of denominators.
Multiplying both sides of the equation by the LCD’s will do the job. Once this is done, the rational
equation should then be polynomial and would be easier to solve. There is a need for the student
to be proficient in factoring and understanding how to get the zeros of polynomial functions. For
this, the teacher might need to reintroduce the rational zeros theorem, and to some extent if
necessary, the Descartes’ Rule of Signs.
𝒙−𝟏=
𝒙 + 𝟒𝟑
𝒙−𝟕
𝒙𝟐
𝟒𝟐
=𝒙
+ 𝟐𝒙 − 𝟐𝟗
(𝑥 − 1)(𝑥 − 7) = 𝑥 + 43
42 = 𝑥(𝑥 2 + 2𝑥 − 29)
𝑥 2 − 8𝑥 + 7 = 𝑥 + 43
𝑥 3 + 2𝑥 2 − 29𝑥 + 42 = 0
𝑥 2 − 9𝑥 − 36 = 0
(𝑥 − 3)(𝑥 + 7)(𝑥 − 2) = 0
(𝑥 − 12)(𝑥 + 3) = 0
𝑥 − 3 = 0; 𝑥 + 7 = 0; 𝑥 − 2 = 0
𝑥 − 12 = 0; 𝑥 + 3 = 0
𝒙 = 𝟑; 𝒙 = −𝟕; 𝒙 = 𝟐
𝒙 = 𝟏𝟐; 𝒙 = −𝟑
Rational Equations and Inequalities  19
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
2𝑥 3 − 𝑥 2 − 𝑥 = 0
𝒙𝟑 − 𝟐𝒙 + 𝟏 𝟏
= 𝒙−𝟏
𝒙−𝟏
𝟐
𝑥(2𝑥 + 1)(𝑥 − 1) = 0
𝑥 3 − 2𝑥 + 1
1
(
) ∙ 2(𝑥 − 1) = ( 𝑥 − 1) ∙ 2(𝑥 − 1)
𝑥−1
2
𝑥 = 0; 2𝑥 + 1 = 0; 𝑥 − 1 = 0
𝟏
𝑥 = 𝟎; 𝒙 = − ; 𝒙 = 𝟏
𝟐
2𝑥 3 − 4𝑥 + 2 = 𝑥 2 − 3𝑥 + 2
When dealing with inequalities, it would be unwise to multiply both sides with the LCD as we don’t
know exactly if the expression is positive or negative. If it is negative, the inequality symbol would
have to change. Therefore, the best way to solve a rational iunequality is to transpose all terms on
the left side of the equation and simplify it there by unifying the fraction.
𝟐𝒙 + 𝟏𝟕
𝒙𝟑 − 𝟐𝒙 + 𝟏 𝟏
For the succeeding three
> 𝒙+𝟓
≥ 𝒙−𝟏
3
2
𝒙+𝟏
𝒙−𝟏
𝟐
examples on solving rational
3
2𝑥
+
17
inequalities, it is important to
𝑥 − 2𝑥 + 1 1
− (𝑥 + 5) > 0
− 𝑥+1≥0
𝑥+1
explain comprehensively
𝑥−1
2
3
2𝑥 + 17 (𝑥 + 5)(𝑥 + 1)
the need for sign diagrams.
2(𝑥
−
2𝑥
+
1)
−
𝑥(𝑥
−
1) + 2(𝑥 − 1)
−
>0
≥0
𝑥+1
𝑥+1
The process of constructing
2(𝑥 − 1)
one is explained under
2𝑥 + 17 𝑥 2 + 6𝑥 + 5
2𝑥 3 − 𝑥 2 − 𝑥
−
>0
≥0
“Essential Ideas.”
𝑥+1
𝑥+1
2𝑥 − 2
−𝑥 2 − 4𝑥 + 12
2𝑥 3 − 𝑥 2 − 𝑥 = 0
>0
𝒙−𝟑
𝑥+1
≤𝟎
1
𝑥(2𝑥 2 − 𝑥 − 1) = 0
𝒙+𝟐
𝑥 2 + 4𝑥 − 12
<0
𝑥(2𝑥 + 1)(𝑥 − 1) = 0
𝐼𝑓 𝒙 = 𝟑, 𝑓(𝑥) = 0
𝑥+1
!
(+)
0 (+)
(-)
-2
3
2𝑥 − 2 ≠ 0; 𝒙 ≠ 𝟏
𝑥 2 + 4𝑥 − 12
<0
𝑥+1
𝑥 + 2 ≠ 0; 𝒙 ≠ −𝟐
(𝑥 − 2)(𝑥 + 6) = 0; 𝒙 ≠ 𝟐, −𝟔
𝑥 + 1 ≠ 0; 𝒙 ≠ −𝟏
(−𝟐, 𝟑)
(-) 0
(+)
-6
!
(-)
-1
0 (+)
(+)
0
(-)
-1/2
0
0
(+)
!
(+)
1
𝟏
(−∞, − ] ∪ [𝟎, 𝟏) ∪ (𝟏, +∞)
𝟐
2
(−∞, −𝟔) ∪ (−𝟏, 𝟐)
The following question should be raised to aid generalization of the concept targeted: Briefly, how
would you explain the process of solving rational equations and inequalities?
As exercise, solve the rational equations and inequalities. For the inequalities, express answer in
interval notation.
𝑥
𝟔
a.
=3
[𝑥 = 15𝑥 + 12; −14𝑥 = 12; 𝒙 = − ]
b.
c.
d.
5𝑥+4
3𝑥−1
𝑥 2 +1
1
𝑥+2
𝑥
𝟕
=1
2
[3𝑥 − 1 = 𝑥 + 1; −𝑥 + 3𝑥 − 2 = 0; 𝑥 2 − 3𝑥 + 2 = 0; (𝑥 − 2)(𝑥 − 1) = 0; 𝒙 = 𝟐, 𝟏 ]
≥0
𝑥 2 −1
2
[𝑥 + 2 ≠ 0; 𝑥 ≠ −2 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡, 𝑦 =
>0
1
2
(−𝟐, +∞)]
[𝑥 2 − 1 ≠ 0; 𝑥 ≠ ±1, 𝑧𝑒𝑟𝑜: 𝑥 = 0, 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 𝑦 = 0, (−𝟏, 𝟎) ∪ (𝟏, +∞)]
WRITTEN WORK NO. 4
Assignment (Pen and Paper Test)
Solve the following rational equations and inequalities.
1.
2.
3.
4.
2𝑥−7
𝑥+3
=2
1
𝑥 2 −3
1
+ 𝑥−3 = 𝑥 2 −9
𝑥+3
𝑥 2 −𝑥−12
𝑥 2 +𝑥−6
>0
3𝑥 2 −5𝑥−2
𝑥 2 −9
<0
Highest Possible Score: 20 points
Passing Score: 15 points (75 percent)
Rational Equations and Inequalities  20
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Day Two [Target M11GM-Ic-3]
Solve problems involving rational equations and inequalities
PROBLEM SOLVING
1. Do routines and other preparatory activities for five minutes. Collect the assignment and discuss
items that need to be discussed.
2. Present the lesson and the targeted competency.
3. Facilitate activation of prior knowledge by using the following questions for recapitulation.
a. How would we differentiate rational equations from rational inequalities?
b. Wxplain the technique in solving rational equations and inequalities.
4. Discuss the following problems involving rational equations and inequalitites to the class.
Given a cost function 𝑪(𝒙), which returns the total cost of producing x items, the average cost function,
̅(𝒙) = 𝑪(𝒙) computes the cost per item when 𝒙 items are produced. Suppose the cost 𝑪, in pesos, to
𝑪
𝒙
produce 𝒙 cellphone protective cases for a local retailer is 𝑪(𝒙) = 𝟖𝟎𝒙 + 𝟏𝟓𝟎, 𝒙 ≥ 𝟎.
Find an expression for the
Solve 𝐶̅ (𝑥) < 100 and interpret.
Determine the behavior of 𝐶̅ (𝑥)as
80𝑥 + 150
average cost function, 𝐶̅ (𝑥).
𝑥 → ∞ and interpret.
< 100
𝟖𝟎𝒙 + 𝟏𝟓𝟎
̅ (𝒙)
𝑥
𝑪
𝑥
̅(𝒙) =
𝑪
,𝒙 > 𝟎
80𝑥
+
150
1
230
𝒙
− 100 < 0
𝑥
10
95
80𝑥 + 150 − 100
100
81.50
<0
𝑥
1000
80.15
−20𝑥 + 150
<0
10000
80.015
𝑥
100000 80.0015
𝑥 > 0, 𝑧𝑒𝑟𝑜: 𝑥 = 7.5
As 𝑥 → ∞, 𝐶̅ (𝑥) gets closer and
( 7.5, ∞)
closer to 80. This means that the
In the context of the problem,
average cost per case is always
solving 𝐶̅ (𝑥) < 100 means we are
greater than PhP 80.00 but is
trying to find how many systems we
need to produce so that the average approaching this amount as more
cost is less than PhP 100.00 per case. and more cases are produced.
Our solution tells us that we need to
produce more than 7.5 cases to
achieve this but it doesn’t make
1
sense to produce just of a case so
2
our final answer should be 8.
A box with a square base and no top is to be constructed so that it has a volume of 1000 cubic centimeters.
Let 𝒙 denote the width of the box in centimeters.
Express the height ℎ in
Solve ℎ(𝑥) ≥ 𝑥 and interpret.
Express the surface area 𝑆 of the box
centimeters as a function of
as a function of 𝑥 and state the
1000
1000
the width 𝑥 and state the
applied domain.
≥ 𝑥;
−𝑥 ≥0
applied domain.
𝑥2
𝑥2
1000 − 𝑥 3
1000
≥0
𝑆(𝑥) = 𝑥 2 + 4 (
)
2
The formula for the volume of
𝑥
𝑥
2
4000
𝑥 ≠ 0; 𝑥 ≠ 0
a rectangular prism such as
𝑆(𝑥) = 𝑥 2 +
Solving
for
the
zero
of
the
function,
𝑥
the box is 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙𝑒𝑛𝑔𝑡ℎ×
1000 − 𝑥 3
𝑤𝑖𝑑𝑡ℎ×ℎ𝑒𝑖𝑔ℎ𝑡. With both the
=0
Domain: (0, ∞)
𝑥2
volume and the width given,
3
1000
−
𝑥
=
0
the equation is,
𝑥 3 = 1000
1000 = 𝑥 2 ℎ
𝑥 = 10
1000
ℎ(𝑥) = 2
(−∞, 0) ∪ (0,10)
Solution
set:
𝑥
Therefore the domain of the
function is 𝑥 ≠ 0 and it should But since 𝑥 represents
measurement, it cannot take a
also be greater than zero.
negative quantity, (0,10)
(0, ∞)
This means that the width of the box
can take any measurement as long as
it is less than 10.
Rational Equations and Inequalities  21
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
The following questions should be raised to facilitate abstraction: (a) What challenges do you think
would you have whenever you encounter word problems involving rational functions? (b) What
can be done to minimize the difficulty these challenges impose?
A television set costs PhP 27,400.00 and a yearly cost of electricity of PhP 550.00.
a. Determine the total annual cost for a television set that lasts for 10 years. Assume that the
cost includes electricity and depreciation. (𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 =
𝑐𝑜𝑠𝑡 𝑜𝑓 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒
𝑢𝑠𝑒𝑓𝑢𝑙 𝑙𝑖𝑓𝑒 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠
)
b. Write a function that gives the annual cost of a television set as a function of the number of
years.
c. Determine the asymptotes of the function. Explain the meaning of the horizontal
asymptote in terms of the television set.
Evaluation/Assignment (Performance Task)
In groups of three, students will need to either find a problem related to rational equations and
inequalities or concoct one of their own. They will also need to solve this problem and have their
solutions submitted for evaluation. In order to check that the solution to the selected problem is
not available from the same source as the problem, references need to be indicated.
Use the rubric below as guide in evaluating students’ outputs.
PERFORMANCE TASK NO. 5
Criteria
Can do better
Level of Difficulty
Problem selected was
easy (1 points)
Clarity,
completeness, and
conciseness of
problem statement
The problem lacks some
important information (1
point)
Student’s
understanding of
the problem
Translation and
representation of
variables is incorrect (1
point)
Correctness of
solution presented
The solution contains at
least two errors (3 point)
Group dynamics
Members worked
independently or more
than one did not help with
the task at hand. (1 point)
Time Management
Task completed past the
intended time. (no credit)
Score Interpretation
10 points or less
Predictors
Doing okay
Meeting expectations
Problem selected was
average in terms of
difficulty (2 points)
All information needed to
solve the problem is
present but the manner
by which it was stated
may be redundant or even
misleading (2 points)
Problem selected was
difficult (3 points)
Problem statement is
short and clear (3 points)
Mathematical sentence
constructed from the
problem is correct (3
points)
The solution contains one
error (4 points)
The solution is correct (5
points)
One member clearly did
not participate. (2 points)
The project reflects
collaboration among
members of the group. (3
points)
Task completed on or
before the intended time
(3 points)
11 to 16 points
17 to 20 points
Make sure to return the outputs with the appropriate notations.
Rational Equations and Inequalities  22
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Inverse Functions
Week Four, Functions and Their Graphs
Content Standards: The learner demonstrates understanding of key
concepts of inverse functions, exponential functions, and
logarithmic functions.
Performance Standards: The learner is able to accurately apply the
concepts of inverse functions, exponential functions, and
logarithmic functions to formulate and solve real-life problems with
precision and accuracy.
Learning Competencies: Over the course of one week, the learner is
expected to (1) represent real-life situations using one-to-one
functions [M11GM-Id-1]; (2) determine the inverse of a one-to-one
function [M11GM-Id-2]; (3) represent an inverse function through its
(a) table of values, and (b) graph [M11GM-Id-3]; (4) find the
domain and range of inverse functions [M11GM-Id-4]; and (5)
graphs inverse functions [M11GM-Ie-1]
Learning Materials: In order to develop the targeted competencies,
the following materials are needed: (a) chalkboard and chalk; (b)
LCD projector and laptop or in the absence, visual materials; (c)
calculators; (d) worksheets
Expected Outputs: In order to assess the attainment of the learning
competencies targeted, students will be required to undertake two
group performance tasks and one written work.
Procedure (Teacher’s Activity)
ONE-TO-ONE FUNCTIONS AND
Day One [Target M11GM-Id-1 and
THEIR INVERSE
M11GM-1d-2]
Represent real-life situations using one-to-one functions and
determine the inverse of a one-to-one function.
1. Do routines and other preparatory activities for five minutes.
2. Present the lesson and the targeted competencies (and other
learning outcome).
3. Facilitate recall with the following questions:
a. What are relations?
b. What are the three possible correspondences of ordered
pairs for relations?
c. Which of these correspondences are functions? Explain
Essential Ideas
Stick-to-one
•••
A function f is said to be oneto-one if f matches different
inputs to different outputs.
Equivalently, f is one-to-one
if and only if whenever
𝑓(𝑐) = 𝑓(𝑑), then 𝑐 = 𝑑.
Horizontal Line Test:
A function f is said to be oneto-one if and only if no
horizontal line intersects the
graph of f more than once.
Suppose f and g are two
functions such that
(1) (𝑔 ∘ 𝑓)(𝑥) = 𝑥 for all 𝑥 in
the domain of f and
(2) (𝑓 ∘ 𝑔)(𝑥) = 𝑥 for all 𝑥 in
the domain of g
then f and g are inverses of
each other and the functions f
and g are said to be invertible.
Properties of Inverse
Functions:
• The range of f is the domain
of g and the domain of f is
the range of g.
• 𝑓(𝑎) = 𝑏 if and only if
𝑔(𝑏) = 𝑎
• (𝑎, 𝑏) is on the graph of f if
and only if (𝑏, 𝑎) is on the
graph of g
Uniqueness of Inverse
Functions:
There is exactly one inverse
function for f denoted by 𝑓 −1
(read as f-inverse)
4. Ask the class to determine which of the following functions are one-to-one. If a function is not oneto-one, they need to provide two ordered pairs that belong to the function but share ordinates.
(1)𝑓(𝑥) = 3𝑥 + 4
(2) 𝑔(𝑥) = 𝑥 2
(3) ℎ(𝑥) = 2𝑥 3
2
(5)𝑔(𝑥) = 4
(4) 𝑓(𝑥) = 7 − 2𝑥
(6) 𝑚(𝑥) = |𝑥 − 1|
𝑥
(7)
(8)
Inverse Functions  23
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Answers to the exploration activity in the previous page: (1) one-to-one; (2) No. (−2,4)𝑎𝑛𝑑 (2,4); (3) one-toone; (4) one-to-one; (5) No. (−1,2)𝑎𝑛𝑑 (1,2); (6) No. (−2,3)𝑎𝑛𝑑 (4,3); (7) No. (−1,3)𝑎𝑛𝑑 (7,3); (8) one-to-one
5. Consider two of the functions from the previous activity, specifically, 𝑓(𝑥) = 3𝑥 + 4 and 𝑔(𝑥) = 𝑥 2 .
Allow the students to construct tables of values for both functions, such as the ones that follow.
𝑓(𝑥) = 3𝑥 + 4
𝑔(𝑥) = 𝑥 2
𝑥
𝑥
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
𝑓(𝑥)
𝑔(𝑥)
-5
-2
1
4
7
10
13
9
4
1
0
1
4
9
Ask one student to remind the class why the two are functions?
[No two ordered pairs share the same domain]
Teacher: Let us say we want to interchange the values of the dependent and the independent variables in both
functions we have just constructed tables for.
𝑓′(𝑥) =____________
𝑔′(𝑥) = _____________
𝑥
𝑥
-5
-2
1
4
7
10
13
9
4
1
0
1
4
9
𝑓(𝑥)
𝑔(𝑥)
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
Teacher: What can be said about the resulting tables?
[Interchanging the values in 𝑓(𝑥) resulted to another function which cannot be said (is not the case) for 𝑔(𝑥)]
Teacher: Who can tell us what expressions represent the ordered pairs in the last two tables?
[𝑓′(𝑥) =
𝑥−4
3
𝑔′ (𝑥) = √𝑥]
Teacher: The last two relations are the inverses of the functions we started with. Note that not all functions have
a function for its inverse as in the case of 𝑔(𝑥). Only one-to-one functions have inverses that are also functions.
These are called invertible.
6. How do we obtain the inverse of a one-to-one function? For simple functions such as 𝑓(𝑥) = 3𝑥 + 4,
working backwards is one technique. For example, if 𝑥 = 2, to get the value of 𝑓(2), we multiply by
3 first then add to 4 to get an answer of 10.
Starting with 10 in order to obtain 2, we subtract by 4 then divide by 3 instead, hence, 𝑓 −1 (𝑥) =
𝑥−4
3
.
Teacher: Can you do the same with 𝑓(𝑥) = −2𝑥 + 7?
[𝑓 −1 (𝑥) =
𝑥−7
−2
or 𝑓 −1 (𝑥) =
7−𝑥
2
]
7. Discuss the second, more procedural method for getting the inverse of a function with the two
examples below.
Example 1: 𝒇(𝒙) = 𝟖𝒙 − 𝟑
Procedure
𝑦 = 8𝑥 − 3
Temporarily replace the function
notation by the dependent
variable 𝑦
𝑥 = 8𝑦 − 3
Interchange 𝑥 and 𝑦
𝑥 = 8𝑦 − 3
−8𝑦 = −𝑥 − 3
𝑥+3
𝑦=
8
Solve for 𝑦 in terms of 𝑥.
𝒇−𝟏 (𝒙) =
𝒙+𝟑
𝟖
Revert back to function notation,
this time, using the notation,
𝑓 −1 (𝑥).
Example 2: 𝒈(𝒙) =
𝑦=
𝟔
𝒙+𝟑
6
𝑥+3
6
𝑦+3
6
(𝑦 + 3) (𝑥 =
) (𝑦 + 3)
𝑦+3
𝑥𝑦 + 3𝑥 = 6
𝑥𝑦 = −3𝑥 + 6
−3𝑥 + 6
𝑦=
𝑥
𝑥=
𝒈−𝟏 (𝒙) =
−𝟑𝒙 + 𝟔
𝒙
The following questions should be raised to aid generalization of the concept targeted: What are
one-to-one functions? Inverse functions? How do we determine the inverse function of a one-toone function?
Inverse Functions  24
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
As exercise, students need to determine the inverse function of the following one-to-one functions.
1−2𝑥
1−5𝑥
a. 𝑓(𝑥) =
[𝑓 −1 (𝑥) =
]
5
b. 𝑚(𝑥) = 1 −
c. 𝑘(𝑥) =
2𝑥−1
3𝑥+4
2
4+3𝑥
5
[𝑚−1 (𝑥) = −
[𝑘 −1 (𝑥) = −
4𝑥+1
3𝑥−2
5𝑥−1
3
]
]
Assignment. Research and then describe ways by which we can prove that two known functions
1
are inverses of each other. Use the function, ℎ(𝑥) = √3𝑥 − 1 + 5 and its inverse, ℎ−1 (𝑥) = (𝑥 − 5)2 +
3
1
3
, 𝑥 ≥ 5. Then, provide another example (a function and its inverse) to further demonstrate the
method(s) you discussed.
PERFORMANCE TASK NO. 6
Criteria
Generalizability of
the method(s)
presented
Clarity of
explanation/
description
Application of the
method to the
enforced example
Application of the
method to the
student-made
example
More than one
correct method is
presented
References are
properly cited
Score Interpretation
Predictors
Doing okay
Can do better
The method presented
does not apply to all
inverse functions (3
points)
The description lacks
some important
information and is
generally unclear (3 point)
The process is entirely
incorrect but a solution I
at least provided following
the suggested method (1
point)
The functions provided
are not even correct
inverses of each other (1
point)
Meeting expectations
The method presented
applies to all inverse
functions (5 points)
Description is stated
clearly (5 points)
There are at most two
errors in the process
following the suggested
method (2 point)
The solution is flawless (3
points)
The functions are inverses
but the solution contains
errors (2 points)
The solution is flawless (3
points)
(additional 1 point)
(additional 1 point)
8 points or less
9 to 14 points
15 to 18 points
REPRESENTING INVERSE FUNCTIONS
Day Two [Target M11GM-Id-3 and M11GM-Id-4]
Represent an inverse function through its (a) table of values, and (b) graph; and find the domain and
range of inverse functions.
1. Do routines and other preparatory activities for five minutes. Collect the assignment and discuss
the item if it is deemed necessary.
The purpose of the assignment was for students to find out through individual research that using
composition of functions, we can verify if two functions are actually inverses of each other. Expect
that some students were unable to dig this up in their assignment so have those who found out
discuss the process in class. The correct solution is given as follows:
1
1
The functions ℎ(𝑥) = √3𝑥 − 1 + 5 and ℎ−1 (𝑥) = (𝑥 − 5)2 + , 𝑥 ≥ 5 are inverses of each other if and only if
3
3
(ℎ ∘ ℎ−1 ) = (ℎ−1 ∘ ℎ)(𝑥) = 𝑥. Hence, to check,
(ℎ ∘ ℎ−1 )(𝑥) = ℎ(ℎ−1 (𝑥))
(ℎ−1 ∘ ℎ)(𝑥) = ℎ−1 (ℎ(𝑥))
1
1
2
1
1
ℎ−1 (ℎ(𝑥)) = (√3𝑥 − 1 + 5 − 5) +
−1 (𝑥))
2
3
3
ℎ(ℎ
= √3 ( (𝑥 − 5) + ) − 1 + 5
3
3
Inverse Functions  25
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
1
1
2
(√3𝑥 − 1) +
3
3
1
1
−1
ℎ (ℎ(𝑥)) = (3𝑥 − 1) +
3
3
1 1
−1
ℎ (ℎ(𝑥)) = 𝑥 − +
3 3
ℎ−1 (ℎ(𝑥)) = 𝑥
ℎ(ℎ−1 (𝑥)) = √(𝑥 − 5)2 + 1 − 1 + 5
ℎ(ℎ−1 (𝑥)) = √(𝑥 − 5)2 + 5
ℎ(ℎ−1 (𝑥)) = (𝑥 − 5) + 5
ℎ(ℎ−1 (𝑥)) = 𝑥
ℎ−1 (ℎ(𝑥)) =
The solutions in the previous page show that the two functions are inverses of each other.
2. Present the lesson and the targeted competency.
3. Facilitate activation of prior knowledge by using the following questions for recapitulation.
a. What are one-to-one functions?
b. What are inverse functions? Do all functions have inverse functions?
c. How do we determine the inverse function of a one-to-one function?
4. Divide the class into ten groups for this topic’s activity. Each
group shall be assigned to just one function and must
accomplish the following: (a) determine the inverse function of
the function they are assigned to; (b) construct a table of values
composed of four ordered pairs for their function; and(c) graph
he function they were assigned to.
For the purpose of presentation, students will need the graph
their function on a meter’s length of acetate (all groups should
have the same size) the teacher shall provide a standard
measure of the Cartesian plane on a manila paper.
acetate over manila paper
When graphing students needs to place their acetate above
this manila paper and plot their points, sketch their graph using
permanent markers. Ask the groups working with the first five
functions to use blue ink and red for the other five. The reason
for this is the graphs need to be compared by putting one
graph over the other after they have been individually
presented.
The assigning of functions may be done at random through
fishbowl. These functions are the following:
𝑓(𝑥) = 3𝑥 + 1
𝑓(𝑥) = 2𝑥 − 3
𝑓(𝑥) = 𝑥 2 , 𝑥 > 0
𝑥−1
𝑥+3
𝑓(𝑥) = 𝑥 + 4
𝑓(𝑥) = √𝑥
𝑓(𝑥) =
𝑓(𝑥) =
3
2
A maximum of 5 minutes may be alloted for the undertaking of this task.
𝑓(𝑥) = 𝑥 − 4
𝑓(𝑥) = 𝑥 3
3
𝑓(𝑥) = √𝑥
Objective: To show and conclude that the graphs of two inverse functions are symmetric with respect to the
graph of the identity function, 𝒚 = 𝒙.
Function,
𝒇(𝒙)
𝑥−4
Inverse
Function
𝒇−𝟏 (𝒙)
𝑥+4
Table of Values
𝑥
𝑓(𝑥)
𝑥
𝑓 −1 (𝑥)
-1
-5
-5
-1
0
-4
-4
0
1
-3
-3
1
Graph (𝒇(𝒙); 𝒇−𝟏 (𝒙); 𝒚 = 𝒙)
2
-2
-2
2
Inverse Functions  26
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
3𝑥 + 1
𝑥−1
3
𝑥
𝑓(𝑥)
𝑥
𝑓 −1 (𝑥)
-1
-2
-2
-1
0
1
1
0
1
4
4
1
2
7
7
2
2𝑥 − 3
𝑥+3
2
𝑥
𝑓(𝑥)
𝑥
𝑓 −1 (𝑥)
-1
-5
-5
-1
0
-3
-3
0
1
-1
-1
1
2
1
1
2
√𝑥
𝑥
𝑓(𝑥)
𝑥
𝑓 −1 (𝑥)
0
0
0
0
1
1
1
1
2
4
4
2
3
9
9
3
𝑥
𝑓(𝑥)
𝑥
𝑓 −1 (𝑥)
-1
-1
-1
-1
0
0
0
0
1
1
1
1
2
8
8
2
𝑥 2, 𝑥 > 0
𝑥
3
3
√𝑥
Guide Questions (distribute or reveal these questions AFTER the group finished their graphs) Give the groups
another five minutes to discuss answers to the questions below before using them as guide for the analysis of
the activity.
1.
2.
What can be said about the domain and range of a function and its inverse function?
[The domain of a function is the range of its inverse and vice versa]
How is the slope of the graph of a linear function related to the slope of the graph of its inverse?
[The slope of a linear function’s graph is the reciprocal of the slope of the graph of its inverse function]
Inverse Functions  27
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
3.
Did you notice any pattern or relationship between the graphs of each pair of inverse functions? Is it
possible to determine the graph of a function from its inverse? Discuss.
[The graphs of inverse functions are symmetric with respect to the graph of the identity function, 𝒚 = 𝒙]
5. Ask the groups to present their graphs to class. The presentation should only cover how they went
about their table of values and then the graph itself. Each group should only consume 2 minutes.
Then, pair up the inverse functions and compare their graphs by putting one acetate over the
other. Do this for all pairs.
With the aid of the guide questions enumerated above, facilitate a class discussion revolving
around the outputs of the students. Emphasize
the pattern graphs of inverse functions have.
As an aid for abstraction, ask the same questions
used as guide questions in the activity. However,
in this case, the class should have already
agreed on generalized answers to these
questions.
Graph the function, 𝑓(𝑥) = −
𝑥−3
3
and its inverse.
Show that the graphs are symmetric with respect
to the graph of the identity function. Also,
determine the domain and range of these two
functions.
PERFORMANCE TASK NO. 7
Evaluation. Use the rubric below to score students’ output in the activity.
Criteria
Can do better
Predictors
Doing okay
Meeting expectations
Correctness of graph
presented
The graph is incorrect (2
point)
The graph is correct (5
points)
Group dynamics
Members worked
independently or more
than one did not help with
the task at hand. (1 point)
The project reflects
collaboration among
members of the group. (3
points)
Time Management
Task completed past the
intended time. (no credit)
Score Interpretation
3 points or less
One member clearly did
not participate. (2 points)
Task completed on or
before the intended time
(2 points)
4 to 7 points
8 to 10 points
2
Given the function, 𝑓(𝑥) = 𝑥 − 1, determine its inverse function and graph both functions. Show
3
WRITTEN WORK NO. 5
that the two functions are symmetric with respect to the identity function’s graph.
Criteria
Accuracy of each
graph
Can do better
Correctness of the
inverse function
derived.
The graph constructed is
not correct. (1 point each)
At least two ordered pairs
in the table are incorrect
(1 point each)
The inverse function
identified is incorrect (1
point)
Score Interpretation
5 points or less
Correctness of table
of values
Predictors
Doing okay
Meeting expectations
The graph constructed is
correct. (3 points each)
One value in the table is
not correct (2 points each)
All values are correct (3
points each)
The correct inverse
function has been
determined (3 points)
6 to 10 points
11 to 15 points
Inverse Functions  28
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Exponential Functions
Week Five, Functions and Their Graphs
Content Standards: The learner demonstrates understanding of key
concepts of inverse functions, exponential functions, and
logarithmic functions.
Performance Standards: The learner is able to accurately apply the
concepts of inverse functions, exponential functions, and
logarithmic functions to formulate and solve real-life problems with
precision and accuracy.
Learning Competencies: Over the course of one week, the learner is
expected to (1) represent real-life situations using exponential
functions [M11GM-Id-1]; (2) determine the inverse of a one-to-one
function [M11GM-Id-2]; (3) represent an inverse function through its
(a) table of values, and (b) graph [M11GM-Id-3]; (4) find the
domain and range of inverse functions [M11GM-Id-4]; and (5)
graphs inverse functions [M11GM-Ie-1]
Learning Materials: In order to develop the targeted competencies,
the following materials are needed: (a) chalkboard and chalk; (b)
LCD projector and laptop or in the absence, visual materials; (c)
calculators; (d) worksheets
Expected Outputs: In order to assess the attainment of the learning
competencies targeted, students will be required to undertake two
group performance tasks and one written work.
Procedure (Teacher’s Activity)
ONE-TO-ONE FUNCTIONS AND
Day One [Target M11GM-Id-1 and
THEIR INVERSE
M11GM-1d-2]
Represent real-life situations using one-to-one functions and
determine the inverse of a one-to-one function.
8. Do routines and other preparatory activities for five minutes.
9. Present the lesson and the targeted competencies (and other
learning outcome).
10. Facilitate recall with the following questions:
d. What are relations?
e. What are the three possible correspondences of ordered
pairs for relations?
f. Which of these correspondences are functions? Explain
Essential Ideas
Stick-to-one
•••
A function f is said to be oneto-one if f matches different
inputs to different outputs.
Equivalently, f is one-to-one
if and only if whenever
𝑓(𝑐) = 𝑓(𝑑), then 𝑐 = 𝑑.
Horizontal Line Test:
A function f is said to be oneto-one if and only if no
horizontal line intersects the
graph of f more than once.
Suppose f and g are two
functions such that
(1) (𝑔 ∘ 𝑓)(𝑥) = 𝑥 for all 𝑥 in
the domain of f and
(2) (𝑓 ∘ 𝑔)(𝑥) = 𝑥 for all 𝑥 in
the domain of g
then f and g are inverses of
each other and the functions f
and g are said to be invertible.
Properties of Inverse
Functions:
• The range of f is the domain
of g and the domain of f is
the range of g.
• 𝑓(𝑎) = 𝑏 if and only if
𝑔(𝑏) = 𝑎
• (𝑎, 𝑏) is on the graph of f if
and only if (𝑏, 𝑎) is on the
graph of g
Uniqueness of Inverse
Functions:
There is exactly one inverse
function for f denoted by 𝑓 −1
(read as f-inverse)
11. Ask the class to determine which of the following functions are one-to-one. If a function is not oneto-one, they need to provide two ordered pairs that belong to the function but share ordinates.
(1)𝑓(𝑥) = 3𝑥 + 4
(2) 𝑔(𝑥) = 𝑥 2
(3) ℎ(𝑥) = 2𝑥 3
2
(5)𝑔(𝑥) = 4
(4) 𝑓(𝑥) = 7 − 2𝑥
(6) 𝑚(𝑥) = |𝑥 − 1|
𝑥
(7)
(8)
Exponential Functions  29
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Answers to the exploration activity in the previous page: (1) one-to-one; (2) No. (−2,4)𝑎𝑛𝑑 (2,4); (3) one-toone; (4) one-to-one; (5) No. (−1,2)𝑎𝑛𝑑 (1,2); (6) No. (−2,3)𝑎𝑛𝑑 (4,3); (7) No. (−1,3)𝑎𝑛𝑑 (7,3); (8) one-to-one
12. Consider two of the functions from the previous activity, specifically, 𝑓(𝑥) = 3𝑥 + 4 and 𝑔(𝑥) = 𝑥 2 .
Allow the students to construct tables of values for both functions, such as the ones that follow.
𝑓(𝑥) = 3𝑥 + 4
𝑔(𝑥) = 𝑥 2
𝑥
𝑥
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
𝑓(𝑥)
𝑔(𝑥)
-5
-2
1
4
7
10
13
9
4
1
0
1
4
9
Ask one student to remind the class why the two are functions?
[No two ordered pairs share the same domain]
Teacher: Let us say we want to interchange the values of the dependent and the independent variables in both
functions we have just constructed tables for.
𝑓′(𝑥) =____________
𝑔′(𝑥) = _____________
𝑥
𝑥
-5
-2
1
4
7
10
13
9
4
1
0
1
4
9
𝑓(𝑥)
𝑔(𝑥)
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
Teacher: What can be said about the resulting tables?
[Interchanging the values in 𝑓(𝑥) resulted to another function which cannot be said (is not the case) for 𝑔(𝑥)]
Teacher: Who can tell us what expressions represent the ordered pairs in the last two tables?
[𝑓′(𝑥) =
𝑥−4
3
𝑔′ (𝑥) = √𝑥]
Teacher: The last two relations are the inverses of the functions we started with. Note that not all functions have
a function for its inverse as in the case of 𝑔(𝑥). Only one-to-one functions have inverses that are also functions.
These are called invertible.
13. How do we obtain the inverse of a one-to-one function? For simple functions such as 𝑓(𝑥) = 3𝑥 + 4,
working backwards is one technique. For example, if 𝑥 = 2, to get the value of 𝑓(2), we multiply by
3 first then add to 4 to get an answer of 10.
Starting with 10 in order to obtain 2, we subtract by 4 then divide by 3 instead, hence, 𝑓 −1 (𝑥) =
𝑥−4
3
.
Teacher: Can you do the same with 𝑓(𝑥) = −2𝑥 + 7?
[𝑓 −1 (𝑥) =
𝑥−7
−2
or 𝑓 −1 (𝑥) =
7−𝑥
2
]
14. Discuss the second, more procedural method for getting the inverse of a function with the two
examples below.
Example 1: 𝒇(𝒙) = 𝟖𝒙 − 𝟑
Procedure
𝑦 = 8𝑥 − 3
Temporarily replace the function
notation by the dependent
variable 𝑦
𝑥 = 8𝑦 − 3
Interchange 𝑥 and 𝑦
𝑥 = 8𝑦 − 3
−8𝑦 = −𝑥 − 3
𝑥+3
𝑦=
8
Solve for 𝑦 in terms of 𝑥.
𝒇−𝟏 (𝒙) =
𝒙+𝟑
𝟖
Revert back to function notation,
this time, using the notation,
𝑓 −1 (𝑥).
Example 2: 𝒈(𝒙) =
𝑦=
𝟔
𝒙+𝟑
6
𝑥+3
6
𝑦+3
6
(𝑦 + 3) (𝑥 =
) (𝑦 + 3)
𝑦+3
𝑥𝑦 + 3𝑥 = 6
𝑥𝑦 = −3𝑥 + 6
−3𝑥 + 6
𝑦=
𝑥
𝑥=
𝒈−𝟏 (𝒙) =
−𝟑𝒙 + 𝟔
𝒙
The following questions should be raised to aid generalization of the concept targeted: What are
one-to-one functions? Inverse functions? How do we determine the inverse function of a one-toone function?
Exponential Functions  30
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
As exercise, students need to determine the inverse function of the following one-to-one functions.
1−2𝑥
1−5𝑥
d. 𝑓(𝑥) =
[𝑓 −1 (𝑥) =
]
5
e. 𝑚(𝑥) = 1 −
f.
𝑘(𝑥) =
2𝑥−1
3𝑥+4
2
4+3𝑥
5
[𝑚−1 (𝑥) = −
[𝑘 −1 (𝑥) = −
4𝑥+1
3𝑥−2
5𝑥−1
3
]
]
Assignment. Research and then describe ways by which we can prove that two known functions
1
are inverses of each other. Use the function, ℎ(𝑥) = √3𝑥 − 1 + 5 and its inverse, ℎ−1 (𝑥) = (𝑥 − 5)2 +
3
1
3
, 𝑥 ≥ 5. Then, provide another example (a function and its inverse) to further demonstrate the
method(s) you discussed.
PERFORMANCE TASK NO. 6
Criteria
Generalizability of
the method(s)
presented
Clarity of
explanation/
description
Application of the
method to the
enforced example
Application of the
method to the
student-made
example
More than one
correct method is
presented
References are
properly cited
Score Interpretation
Can do better
The method presented
does not apply to all
inverse functions (3
points)
The description lacks
some important
information and is
generally unclear (3 point)
The process is entirely
incorrect but a solution I
at least provided following
the suggested method (1
point)
The functions provided
are not even correct
inverses of each other (1
point)
Predictors
Doing okay
Meeting expectations
The method presented
applies to all inverse
functions (5 points)
Description is stated
clearly (5 points)
There are at most two
errors in the process
following the suggested
method (2 point)
The solution is flawless (3
points)
The functions are inverses
but the solution contains
errors (2 points)
The solution is flawless (3
points)
(additional 1 point)
(additional 1 point)
8 points or less
9 to 14 points
15 to 18 points
REPRESENTING INVERSE FUNCTIONS
Day Two [Target M11GM-Id-3 and M11GM-Id-4]
Represent an inverse function through its (a) table of values, and (b) graph; and find the domain and
range of inverse functions.
6. Do routines and other preparatory activities for five minutes. Collect the assignment and discuss
the item if it is deemed necessary.
Exponential Functions  31
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Exponential Functions
Week Five, Functions and Their Graphs
Content Standards: The learner demonstrates understanding of key
concepts of inverse functions, exponential functions, and
logarithmic functions.
Performance Standards: The learner is able to accurately apply the
concepts of inverse functions, exponential functions, and
logarithmic functions to formulate and solve real-life problems with
precision and accuracy.
Learning Competencies: Over the course of one week, the learner is
expected to (1) represent real-life situations using exponential
functions [M11GM-Ie-3]; (2) distinguishes between exponential
function, exponential equation, and exponential inequality
[M11GM-Ie-4]; (3) solve exponential equations and inequalities
[M11GM-Ie-f-1]; (4) represents an exponential function through its:
(a) table of values; (b) graph; and (c) equation [M11GM-If-2]
Learning Materials: In order to develop the targeted competencies,
the following materials are needed: (a) chalkboard and chalk; (b)
LCD projector and laptop or in the absence, visual materials; (c)
calculators; (d) worksheets
Expected Outputs: In order to assess the attainment of the learning
competencies targeted, students will be required to undertake two
group performance tasks and one written work.
Procedure (Teacher’s Activity)
INTRODUCTION TO EXPONENTIAL
Day One [Target M11GM-Id-1 and
FUNCTIONS
M11GM-1d-2]
Represent real-life situations using one-to-one functions and
determine the inverse of a one-to-one function.
15. Do routines and other preparatory activities for five minutes.
16. Present the lesson and the targeted competencies (and other
learning outcome).
17. Facilitate recall with the following questions:
g. What are relations?
h. What are the three possible correspondences of ordered
pairs for relations?
i. Which of these correspondences are functions? Explain
Essential Ideas
Stick-to-one
•••
A function f is said to be oneto-one if f matches different
inputs to different outputs.
Equivalently, f is one-to-one
if and only if whenever
𝑓(𝑐) = 𝑓(𝑑), then 𝑐 = 𝑑.
Horizontal Line Test:
A function f is said to be oneto-one if and only if no
horizontal line intersects the
graph of f more than once.
Suppose f and g are two
functions such that
(1) (𝑔 ∘ 𝑓)(𝑥) = 𝑥 for all 𝑥 in
the domain of f and
(2) (𝑓 ∘ 𝑔)(𝑥) = 𝑥 for all 𝑥 in
the domain of g
then f and g are inverses of
each other and the functions f
and g are said to be invertible.
Properties of Inverse
Functions:
• The range of f is the domain
of g and the domain of f is
the range of g.
• 𝑓(𝑎) = 𝑏 if and only if
𝑔(𝑏) = 𝑎
• (𝑎, 𝑏) is on the graph of f if
and only if (𝑏, 𝑎) is on the
graph of g
Uniqueness of Inverse
Functions:
There is exactly one inverse
function for f denoted by 𝑓 −1
(read as f-inverse)
18. Ask the class to determine which of the following functions are one-to-one. If a function is not oneto-one, they need to provide two ordered pairs that belong to the function but share ordinates.
(1)𝑓(𝑥) = 3𝑥 + 4
(2) 𝑔(𝑥) = 𝑥 2
(3) ℎ(𝑥) = 2𝑥 3
2
(5)𝑔(𝑥) = 4
(4) 𝑓(𝑥) = 7 − 2𝑥
(6) 𝑚(𝑥) = |𝑥 − 1|
𝑥
(7)
(8)
Exponential Functions  32
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
Answers to the exploration activity in the previous page: (1) one-to-one; (2) No. (−2,4)𝑎𝑛𝑑 (2,4); (3) one-toone; (4) one-to-one; (5) No. (−1,2)𝑎𝑛𝑑 (1,2); (6) No. (−2,3)𝑎𝑛𝑑 (4,3); (7) No. (−1,3)𝑎𝑛𝑑 (7,3); (8) one-to-one
19. Consider two of the functions from the previous activity, specifically, 𝑓(𝑥) = 3𝑥 + 4 and 𝑔(𝑥) = 𝑥 2 .
Allow the students to construct tables of values for both functions, such as the ones that follow.
𝑓(𝑥) = 3𝑥 + 4
𝑔(𝑥) = 𝑥 2
𝑥
𝑥
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
𝑓(𝑥)
𝑔(𝑥)
-5
-2
1
4
7
10
13
9
4
1
0
1
4
9
Ask one student to remind the class why the two are functions?
[No two ordered pairs share the same domain]
Teacher: Let us say we want to interchange the values of the dependent and the independent variables in both
functions we have just constructed tables for.
𝑓′(𝑥) =____________
𝑔′(𝑥) = _____________
𝑥
𝑥
-5
-2
1
4
7
10
13
9
4
1
0
1
4
9
𝑓(𝑥)
𝑔(𝑥)
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
Teacher: What can be said about the resulting tables?
[Interchanging the values in 𝑓(𝑥) resulted to another function which cannot be said (is not the case) for 𝑔(𝑥)]
Teacher: Who can tell us what expressions represent the ordered pairs in the last two tables?
[𝑓′(𝑥) =
𝑥−4
3
𝑔′ (𝑥) = √𝑥]
Teacher: The last two relations are the inverses of the functions we started with. Note that not all functions have
a function for its inverse as in the case of 𝑔(𝑥). Only one-to-one functions have inverses that are also functions.
These are called invertible.
20. How do we obtain the inverse of a one-to-one function? For simple functions such as 𝑓(𝑥) = 3𝑥 + 4,
working backwards is one technique. For example, if 𝑥 = 2, to get the value of 𝑓(2), we multiply by
3 first then add to 4 to get an answer of 10.
Starting with 10 in order to obtain 2, we subtract by 4 then divide by 3 instead, hence, 𝑓 −1 (𝑥) =
𝑥−4
3
.
Teacher: Can you do the same with 𝑓(𝑥) = −2𝑥 + 7?
[𝑓 −1 (𝑥) =
𝑥−7
−2
or 𝑓 −1 (𝑥) =
7−𝑥
2
]
21. Discuss the second, more procedural method for getting the inverse of a function with the two
examples below.
Example 1: 𝒇(𝒙) = 𝟖𝒙 − 𝟑
Procedure
𝑦 = 8𝑥 − 3
Temporarily replace the function
notation by the dependent
variable 𝑦
𝑥 = 8𝑦 − 3
Interchange 𝑥 and 𝑦
𝑥 = 8𝑦 − 3
−8𝑦 = −𝑥 − 3
𝑥+3
𝑦=
8
Solve for 𝑦 in terms of 𝑥.
𝒇−𝟏 (𝒙) =
𝒙+𝟑
𝟖
Revert back to function notation,
this time, using the notation,
𝑓 −1 (𝑥).
Example 2: 𝒈(𝒙) =
𝑦=
𝟔
𝒙+𝟑
6
𝑥+3
6
𝑦+3
6
(𝑦 + 3) (𝑥 =
) (𝑦 + 3)
𝑦+3
𝑥𝑦 + 3𝑥 = 6
𝑥𝑦 = −3𝑥 + 6
−3𝑥 + 6
𝑦=
𝑥
𝑥=
𝒈−𝟏 (𝒙) =
−𝟑𝒙 + 𝟔
𝒙
The following questions should be raised to aid generalization of the concept targeted: What are
one-to-one functions? Inverse functions? How do we determine the inverse function of a one-toone function?
Exponential Functions  33
Von Christopher G. Chua
General Mathematics Teaching Guides
•••
As exercise, students need to determine the inverse function of the following one-to-one functions.
1−2𝑥
1−5𝑥
g. 𝑓(𝑥) =
[𝑓 −1 (𝑥) =
]
5
h. 𝑚(𝑥) = 1 −
i.
𝑘(𝑥) =
2𝑥−1
3𝑥+4
2
4+3𝑥
5
[𝑚−1 (𝑥) = −
[𝑘 −1 (𝑥) = −
4𝑥+1
3𝑥−2
5𝑥−1
3
]
]
Assignment. Research and then describe ways by which we can prove that two known functions
1
are inverses of each other. Use the function, ℎ(𝑥) = √3𝑥 − 1 + 5 and its inverse, ℎ−1 (𝑥) = (𝑥 − 5)2 +
3
1
3
, 𝑥 ≥ 5. Then, provide another example (a function and its inverse) to further demonstrate the
method(s) you discussed.
PERFORMANCE TASK NO. 6
Criteria
Generalizability of
the method(s)
presented
Clarity of
explanation/
description
Application of the
method to the
enforced example
Application of the
method to the
student-made
example
More than one
correct method is
presented
References are
properly cited
Score Interpretation
Can do better
The method presented
does not apply to all
inverse functions (3
points)
The description lacks
some important
information and is
generally unclear (3 point)
The process is entirely
incorrect but a solution I
at least provided following
the suggested method (1
point)
The functions provided
are not even correct
inverses of each other (1
point)
Predictors
Doing okay
Meeting expectations
The method presented
applies to all inverse
functions (5 points)
Description is stated
clearly (5 points)
There are at most two
errors in the process
following the suggested
method (2 point)
The solution is flawless (3
points)
The functions are inverses
but the solution contains
errors (2 points)
The solution is flawless (3
points)
(additional 1 point)
(additional 1 point)
8 points or less
9 to 14 points
15 to 18 points
Exponential Functions  34
Von Christopher G. Chua