M11GM-Q1Module3.pdf

11
General
Mathematics
Quarter 1 – Module 3:
Rational Functions

General Mathematics – Grade 11
Self-Learning Module (SLM)
Quarter 1 – Module 3: Rational Functions
First Edition, 2020
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Development Team of the Module
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Editors: Live C. Angga
Reviewers: Reynaldo C. Tagala, Hannih Lou T. Bantilan
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Gilbert B. Barrera – Chief, CLMD
Arturo D. Tingson Jr. – REPS, LRMS
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Jade T. Palomar – REPS, Mathematics
Donna S. Panes – Chief, CID
Elizabeth G. Torres – EPS, LRMS
Judith B. Alba – EPS, ADM
Reynaldo C. Tagala – EPS, Mathematics

2
Introductory Message
For the facilitator:
Welcome to the Grade 11 General Mathematics Self-Learning Module (SLM) on
Rational Functions.
This module was collaboratively designed, developed and reviewed by educators both
from public and private institutions to assist you, the teacher or facilitator in helping
the learners meet the standards set by the K to 12 Curriculum while overcoming
their personal, social, and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and independent
learning activities at their own pace and time. Furthermore, this also aims to help
learners acquire the needed 21st century skills while taking into consideration their
needs and circumstances.
In addition to the material in the main text, you will also see this box in the body of
the module:
As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to manage
their own learning. Furthermore, you are expected to encourage and assist the
learners as they do the tasks included in the module.
Notes to the Teacher
This contains helpful tips or strategies that
will help you in guiding the learners.

3
For the learner:
Welcome to the Grade 11 General Mathematics Self-Learning Module (SLM) on
Rational Functions.
The hand is one of the most symbolized part of the human body. It is often used to
depict skill, action and purpose. Through our hands we may learn, create and
accomplish. Hence, the hand in this learning resource signifies that you as a learner
is capable and empowered to successfully achieve the relevant competencies and
skills at your own pace and time. Your academic success lies in your own hands!
This module was designed to provide you with fun and meaningful opportunities for
guided and independent learning at your own pace and time. You will be enabled to
process the contents of the learning resource while being an active learner.
This module has the following parts and corresponding icons:
What I Need to Know This will give you an idea of the skills or
competencies you are expected to learn in the
module.
What I Know This part includes an activity that aims to
check what you already know about the
lesson to take. If you get all the answers
correct (100%), you may decide to skip this
module.
What’s In This is a brief drill or review to help you link
the current lesson with the previous one.
What’s New In this portion, the new lesson will be
introduced to you in various ways such as a
story, a song, a poem, a problem opener, an
activity or a situation.
What is It This section provides a brief discussion of the
lesson. This aims to help you discover and
understand new concepts and skills.
What’s More This comprises activities for independent
practice to solidify your understanding and
skills of the topic. You may check the
answers to the exercises using the Answer
Key at the end of the module.
What I Have Learned This includes questions or blank
sentence/paragraph to be filled in to process
what you learned from the lesson.
What I Can Do This section provides an activity which will
help you transfer your new knowledge or skill
into real life situations or concerns.

4
Assessment This is a task which aims to evaluate your
level of mastery in achieving the learning
competency.
Additional Activities In this portion, another activity will be given
to you to enrich your knowledge or skill of the
lesson learned. This also tends retention of
learned concepts.
Answer Key This contains answers to all activities in the
module.
At the end of this module you will also find:
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.
We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it!
References This is a list of all sources used in developing
this module.

5
What I Need to Know
This module was designed and written with you in mind. It is here to help you
master the process in determining the (a) Intercepts, (b) Zeroes and (c) Asymptotes
of Rational Functions. The scope of this module permits it to be used in many
different learning situations. The language used recognizes the diverse vocabulary
level of students. The lessons are arranged to follow the standard sequence of the
course. But the order in which you read them can be changed to correspond with
the textbook you are now using.
The module is composed of two lessons, namely:
 Lesson 1 - The: (a) intercepts; (b) zeroes; and (c) asymptotes of rational
functions
 Lesson 2 - problems involving rational functions, equations, and inequalities
After going through this module, you are expected to:
1. determine the intercepts of rational function (M11GM-1c-2);
2. determine the zeroes of rational function (M11GM-1c-2);
3. determine the asymptotes of rational functions (M11GM-Ic-2)
4. define rational functions, rational equations and rational inequalities; and
5. solve problems involving rational functions, equations and inequalities
(M11GM-1c-3).

6
What I Know
Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.
1. Point/s where the graph of the rational function intersects the x- or y-axis.
a. zero c. intercept
b. domain d. asymptote
2. Zeroes are also known as ___________________________________ of functions.
a. x-intercepts c. roots
b. solutions d. all of the above
3. Given the polynomial function 𝑝(𝑥) = 12 + 4𝑥 − 3𝑥2
− 𝑥3
, what is the degree of
polynomial?
a. 4 b. 3 c.2 d.1
4. The values of x which make the function zero.
a. range c. zeroes
5. The real numbered zeroes are also _____________ of the graph of the function.
a. x-intercepts c. range
b. y-intercepts d. asymptote
6. A function of the form
)
(
)
(
)
(
x
q
x
p
x
f  where )
(x
p and )
(x
q are polynomial
functions and is )
(x
q not the zero polynomial.
a. Rational function c. Quadratic function
b. Exponential function d. Linear function
7. Which of the following is a rational function?
a. 8
1
2



x
x
c.
1
3
2
)
(
2




x
x
x
x
f
b. 2

 x
x d.
4
7
2


x
x
8. If the graph of function either increases or decreases without bound as the x-
values approach a from the right or left, the line is called
a. vertical asymptote c. x - axis
b. horizontal asymptote d. y - axis

7
9. If the degrees of the numerator (n) is less than the degrees of the denominator
(m), y = 0 is
a. vertical asymptote c. zero of the function
b. horizontal asymptote d. y - intercept
For number 10-13, consider the function
2
2
)
(



x
x
x
f with its graph below.
10.The x- and y-intercept is
a. -2 and 1 c. 2 and -1
b. 2 and 1 d. -2 and -1
11.The vertical asymptote is
a. x = -2 c. y = -2
b. x = 2 d. y = 2
12.What is the horizontal asymptote of the given function?
a. y = -1 c. y = 2
b. y = 1 d. y = -2
13.What are the zero/es of given function?
a. x = - 1 c. x = 2
b. y = - 1 d. y = -2
14.The following best describes the function
2
3
1
4
4
)
( 2
2





x
x
x
x
x
f EXCEPT,
a. horizontal asymptote is y = 4
b. the x-intercepts and zeroes are the same
c. x = -1 and x = -2 are vertical asymptotes
d. the degree of numerator (n) is greater than the denominator (m)
15.Let n be the degree of the numerator and m be the degree of the denominator.
If n > m,
a. there is no horizontal asymptote c. horizontal asymptote is
b
a
y 
b. horizontal asymptote is y = 0 d. a and b
Source: https://www.desmos.com/calculator (snipped graph)

8
Lesson
1
Intercepts, Zeroes and
Asymptotes of Rational
Functions
In the previous lesson, we have learned the process of finding the domain and
range of rational functions. Understanding the key concepts of domain and range is
very essential as we proceed to this topic. In this module, we will determine the
intercepts, zeroes and asymptotes of rational functions which are mainly the process
in graphing rational functions. Furthermore, rational expressions and equations are
also useful tools for representing real life situations like in describing distance-speed-
time questions and modelling multi-person work problems. With this, we can be able
to analyze and interpret real life problems involving rational functions algebraically
and graphically.
What’s In
Let us start our study of this module by reviewing first the concepts of finding
the domain and range of rational functions.
Find the domain and range of the function
2
2
)
(



x
x
x
f . Algebraically, we get
the following;
a. The domain of )
(x
f is  
2



 x
x .
Observe that the function is undefined at 2


x . This means that 2


x is not
part of the domain of )
(x
f . In addition, no other values of x will make the
function undefined.
b. The range of )
(x
f is  
1


 y
y .
We should know that the range of the function is the same as the domain of
its inverse. Operationally, we get
1
2
2
)
(
1





x
x
x
f . This means that the
function is undefined in 1

y .
On the other hand, the domain of a rational function includes all real numbers
except those that cause the denominator to equal zero.
What are the values of x that will make the function zero? It is
What is the function value when 0

x ?

9
What’s New
Setting Boundaries
Directions: Locate the points which makes the function
2
2
)
(



x
x
x
f undefined.
Draw a vertical broken line for 2


x and a horizontal broken line for
1

y . What do you observe? What will happen if we extend the graph?
Identify the point/s where the graph of the rational function intersects
the x- or y-axis. What are these points?
The vertical broken line and the horizontal line serves as boundaries to which
a function's graph draws closer without touching it.
As x approaches -2 from the left and from the right, the graph gets closer and
closer to the line x = -2.
As x increases or decreases without bound, f(x) gets closer and closer to 1
.That is the line y=1.
The point where the graph of the rational function intersects the x-axis is (2,0).
This represents the value of x that will make the function zero.
In addition, point (0,-1) intersects the y-axis. This represents the function
value when x = 0.
Source: https://www.desmos.com/calculator
Figure 1
x-axis
y-axis

10
What is It
Intercepts of Rational Function
There are two ways in determining the intercepts of rational functions.
Example 1: Let’s consider the given function
2
2
)
(



x
x
x
f .
a. Algebraically, we do the following.
To determine the x-intercept, let y = 0 To determine the y-intercept, let x = 0
2
2
)
(



x
x
x
f
2
2



x
x
y
2
2
)
(



x
x
x
f
2
0
2
0



y
0
2
2



x
x
2
2



y
2

x 1


y
Therefore, x-intercept is at (2,0) Therefore, the y-intercept is at (0,-1)
b. Graphically, we will just locate points where the graph of the rational
function intersects the x- or y-axis. We have done it earlier in figure 1 so it
would be better if we label it on the graph below.
Definition
An intercept of a rational function is a point where the graph of the
rational function intersects the x- or y-axis.
(2,0)
(0,-1) x-axis
y-axis
y-intercept
x-intercept
Figure 2

11
Zeroes of Rational Function
In this case, the numerator x - 2 in the function
2
2
)
(



x
x
x
f will be zero at
x = 2. Therefore x = 2 is a zero of f(x). Since it is a real zero, it is also an x-intercept.
Finally, we can express our answer as x = 2 or (2, 0).
Asymptotes of Rational Function
Vertical Asymptote
We can find vertical asymptotes of rational function by simply following these
steps.
a. Equating the denominator to zero.
b. Solving for x.
In other words, if
)
(
)
(
)
(
x
q
x
p
x
f  . Then setting 0
)
( 
x
q , will give the vertical
asymptote(s). Thus, in
2
2
)
(



x
x
x
f , vertical asymptote is x = -2.
The vertical broken line you have drawn in figure 1 is the vertical asymptote
which serves as boundaries to which a function's graph draws closer without
touching it.
As x approaches -2 from the left and from the right, the graph gets closer and
closer to the line x = -2.
Moreover, a vertical asymptote represents a value at which a rational function
is undefined. So, the domain defines the vertical asymptote of the function.
Definition
Zeroes are also known as x-intercepts, solutions or roots of functions. The
zeroes of a function are the values of x which make the function zero. The real
numbered zeroes are also x-intercepts of the graph of the function.
Definition
An asymptote is a line or curve to which a function's graph draws closer
without touching it. Functions cannot cross a vertical asymptote, and they
usually approach horizontal asymptotes in their end behavior (i.e. as 

x ).
Definition
The vertical line x = a is a vertical asymptote of a function f if the graph
of f either increases or decreases without bound as the x-values approach a from
the right or left.

12
Horizontal Asymptote
There are three possible conditions in determining horizontal asymptote(s) of a
rational function.
Let n be the degree of the numerator and m be the degree of the denominator:
a. If n < m, the horizontal asymptote is y = 0.
b. If n = m, the horizontal asymptote is
b
a
y  , where a is the leading coefficient
of the numerator and b is the leading coefficient of the denominator.
c. If n > m, there is no horizontal asymptote.
The function
2
2
)
(



x
x
x
f satisfies the second condition. We have
1
1


b
a
y ,
therefore horizontal asymptote is y = 1.
The horizontal broken line you have drawn in figure 1 is the horizontal
asymptote which serves as boundaries to which a function's graph draws closer
without touching it.
As x increases or decreases without bound, f(x) gets closer and closer to.That
is the line y=1.
Moreover, the range defines the horizontal asymptote of the function.
The graph below will show us the intercepts, zeroes and asymptotes of
2
2
)
(



x
x
x
f .
Definition
The horizontal line y = b is a horizontal asymptote of the function f if f(x) gets
closer to b as x increases or decreases without bound ( 

x ).
Figure 3
x-axis
y-axis
(2,0)
(0,-1)
HORIZONTAL ASYMPTOTE, y=1
VERTICAL
ASYMPTOTE,
x=-2

13
Example 2: Determine the (a) intercepts, (b) zeroes, and (c) asymptotes of rational
function
16
4
)
( 2
2



x
x
x
f .
Solutions:
a. intercepts
16
4
)
( 2
2



x
x
x
f
16
4
2
2



x
x
y
16
4
)
( 2
2



x
x
x
f
16
4
2
2



x
x
y
0
16
4
2
2



x
x
16
0
4
0



y
0
4
2


x
16
4



y
0
)
2
)(
2
( 

 x
x
2
,
2 

x
4
1

y
Therefore, x-intercepts are (2,0) ,(-2,0) Therefore, the y-intercept is at (0,
4
1
)
b. zeroes are also x-intercepts so 2
,
2 

x or we may express it as (2,0) ,(-2,0).
c. asymptotes,
vertical asymptotes horizontal asymptotes
)
(
)
(
)
(
x
q
x
p
x
f 
16
4
)
( 2
2



x
x
x
f Since n < m, then 0

y
0
)
( 
x
q 0
16
2


x
0
)
4
)(
4
( 

 x
x
4
,
4 

x
function
1
3
2
2
3
)
( 2




x
x
x
x
f .

14
Solutions:
a. intercepts
1
3
2
2
3
)
( 2




x
x
x
x
f
1
3
2
2
3
2




x
x
x
y
1
3
2
2
3
)
( 2




x
x
x
x
f
1
3
2
2
3
2




x
x
x
y
0
1
3
2
2
3
2




x
x
x
1
0
0
2
0




y
0
2
3 

x
1
2

y
3
2


x 2

y
Therefore, x-intercept is (
3
2
 ,0) Therefore, the y-intercept is at (0,2)
b. zeroes are also x-intercepts so
3
2


x or we may express it as (
3
2
 ,0)
c. asymptotes,
)
(
)
(
)
(
x
q
x
p
x
f 
1
3
2
2
3
)
( 2




x
x
x
x
f Since n = m, then
b
a
y 
0
)
( 
x
q 0
1
3
2 2


 x
x
1
1

y
0
)
1
)(
1
2
( 

 x
x
1
,
2
1



x 1

y
function
4
2
1
)
(
2



x
x
x
f .

15
Solutions:
a. intercepts
4
2
1
)
(
2



x
x
x
f
4
2
1
2



x
x
y
4
2
1
)
(
2



x
x
x
f
4
2
1
2



x
x
y
0
4
2
1
2



x
x
4
0
1
0



y
0
1
2


x
4
1


y
1
2


x
Therefore, no real solutions. Therefore, the y-intercept is at (0,
4
1
)
Note: Not all rational functions have both an x or y intercept. If you cannot find a
real solution, then it does not have that intercept.
b. zeroes are also x-intercepts, so as we can see, no real solution.
c. asymptotes,
)
(
)
(
)
(
x
q
x
p
x
f 
4
2
1
)
(
2



x
x
x
f Since n > m,
0
)
( 
x
q 0
4
2 

x no horizontal asymptote
2

x
Determining the intercepts, zeroes and asymptotes of rational functions are
mainly the process in graphing rational functions. We will not be sketching the graph
as it will be discussed in the next module. However, if graphs are given, we can
determine these values as we have done in the beginning.

16
What’s More
Activity 1. COMPLETE ME
Directions: Complete the table of rational functions with intercepts, zeroes and
asymptotes. You may write your solutions algebraically on a separate sheet of paper.
Rational Function
x-
intercept/s
/Zeroes
(x/z)
y-
intercept/s
(y)
Vertical
Asymptote
(va)
Horizontal
Asymptote
(ha)
1.
5
5
)
(



x
x
x
f
2.
2
3
1
4
4
)
( 2
2





x
x
x
x
x
f
3.
1
3
2
4
3
)
( 2




x
x
x
x
f
Activity 2. LOCATE ME
Directions: Determine the intercepts, zeroes and asymptotes by locating it on the
following graphs.
x-intercept/s: _________________
y-intercept: _________________
zero/es: _________________
vertical asymptote: _________________
horizontal asymptote: _______________
x-intercept/s: _________________
y-intercept: _________________
zero/es: _________________
vertical asymptote: _________________
horizontal asymptote: _______________
A B

17
Lesson
2
Problems Involving Rational
Functions, Equations, and
Inequalities
Some examples of rational expressions:
1.
𝑥2+2𝑥+3
𝑥+1
(It is a ratio of two polynomials)
2.
5
𝑥−3
(The numerator 5 is a polynomial of degree 0)
3.
𝑥2+4𝑥−3
2
(Rational expression which is also a polynomial)
4.
1
𝑥+2
𝑥−2
(The expression is equal to
1
(𝑥+2)(𝑥−2)
so it is a rational expression)
Example 1. Solve for x :
2
𝑥
−
3
2𝑥
=
1
5
Solution. 10𝑥 (
2
𝑥
−
3
2𝑥
) = 10𝑥(
1
5
) Multiply both sides by the LCD 10𝑥.
10𝑥 (
2
𝑥
) − 10𝑥 (
3
2𝑥
) = 10𝑥 (
1
5
) Distribute.
20 − 15 = 2𝑥 Simplify, and then solve.
𝑥 =
5
2
Note: Check your answer by substituting
5
2
for 𝑥 to see if you obtain a true
statement.
Definition: A rational expression can be described as a function where either
the numerator, denominator, or both have a variable on it.
Rational Equation
Definition. A rational equation is an equation involving rational expressions.
To solve a rational equation:
(a) Eliminate denominators by multiplying each term of the equation
by the least common denominator or LCD.
(b) Note that eliminating denominators may introduce extraneous
solutions. Check the solutions of the transformed equations with
the original equation.

18
Example 1. Solve the inequality
3𝑥−10
𝑥−4
> 2.
Solution. Rewrite the inequality as a single rational expression. Bring 2 to the left.
3𝑥−10
𝑥−4
− 2 > 0
3𝑥−10
𝑥−4
− 2(
𝑥−4
𝑥−4
) > 0 Multiply 2 by
𝑥−4
𝑥−4
3𝑥−10−2(𝑥−4)
𝑥−4
> 0 Simplify.
3𝑋−10−2𝑋+8
𝑋−4
> 0
𝑥−2
𝑥−4
> 0
At 𝑥 = 2, we have
0
𝑥−4
> 0, which is 𝑎 = 0 point
At 𝑥 = 4, we have
𝑥−2
0
> 0, which is undefined.
Choose convenient test points in the intervals determined by 2 and 4 to determine
the sign of
𝑥−2
𝑥−4
in these intervals. Construct a table of signs as shown below.
Test Point 𝑥 = 0 𝑥 = 2 𝑥 = 3 𝑥 = 4 𝑥 = 5
x-2 - + +
x-4 - - +
(x-2)(x-4) + - undefined +
Since we are looking for the intervals where the rational expression is positive, we
determine the solution set to be (−∞, 2) ∪ (4, ∞).
Example 2. Solve the inequality
2𝑥
𝑥+1
≥ 1.
Solution. Rewrite the inequality as a single rational expression.
2𝑥
𝑥 + 1
− 1 ≥ 0
2𝑥 − (𝑥 + 1)
𝑥 + 1
≥ 0
𝑥 − 1
𝑥 + 1
≥ 0
The rational expression will be zero for 𝑥 = 1 and undefined for 𝑥 = −1. The value 𝑥 =
1 is included while 𝑥 = −1 is not. Use a shaded circle for 𝑥 = 1 (s solution) and
unshaded circle for 𝑥 = −1 (not a solution).
Rational Inequality
Definition. A rational inequality is an inequality involving rational expressions.

19
Choose convenient test points in the intervals determined by -1 and 1 to determine
the sign of
𝑥−1
𝑥+1
in these intervals. Construct a table of signs as shown below.
𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑥 < −1 −1 < 𝑥 < 1 𝑥 > 1
𝑇𝑒𝑠𝑡 𝑃𝑜𝑖𝑛𝑡 𝑥 = −2 𝑥 = 0 𝑥 = 2
𝑥 − 1 - - +
𝑥 + 1 - + +
𝑥 − 1
𝑥 + 1
+ - +
Since we are looking for the intervals where the rational expression is positive or
zero, we determine the solution to be the set {𝑥 ∈ 𝑅|𝑥 < −1 𝑜𝑟 𝑥 ≥ 1}. It can also be
written using interval notation: (−∞, −1) ∪ [1, ∞).
Rational Function
Example 1. Given 𝑓(𝑥) =
10
𝑥−3
(a) Construct a table of values using the numbers from -2 to 8.
(b) Plot the points in the Cartesian plane and determine whether the points form
a smooth curve or a straight line.
Solution.
(a)
(b) Connecting the points, we get the following graph which forms two
different smooth curves.
Example 2. (It would be introduced in an another way of solving inequalities).
𝑥 -2 -1 0 1 2 3 4 5 6 7 8
f(x) -2 -.25 -3.3 -5 -10 Und. 10 5 3.3 2.5 2
Definition. A rational function is a function of the form 𝑓(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
where 𝑝(𝑥)and
𝑞(𝑥) are polynomial functions and 𝑞(𝑥) is a nonzero polynomial. The domain of
𝑓(𝑥) is all values of 𝑥 where 𝑞(𝑥) ≠ 0.

20
Sketch the graph of 𝑓(𝑥) =
3𝑥+6
𝑥−1
.
Solution. The 𝑦-intercept is 𝑓(0) =
6
−1
= −6 (0, −6)
The 𝑥-intercepts will be, 3𝑥 + 6 = 0
𝑥 = −2 (−2,0)
Now, we need to determine the asymptotes.
For the vertical asymptote, we have𝑥 − 1 = 0. So, 𝑥 = 1.
For the horizontal asymptote, we have, 𝑦 =
3
1
= 3.
Since the 𝑥 and 𝑦 intercepts are already in the left region, we won’t need to get any
points there. That means, we’ll just get a point in the right region. Say, 𝑥 = 2.
𝑓(2) =
3(2)+6
2−1
=
12
1
= 12 (2,12)
Plotting these points on the graph, we get the following.
The definition of rational equations, inequalities and functions are summarized
below.
Rational Equation Rational Inequality Rational Function
Definition An equation involving
rational expressions.
An inequality involving
rational expressions.
A function of the form
𝑓(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
where 𝑝(𝑥)
and 𝑞(𝑥) are
polynomial functions
and 𝑞(𝑥) is a nonzero
function (𝑖. 𝑒. , 𝑞(𝑥) ≠ 0).
Example 2
𝑥
−
3
2𝑥
=
1
5
5
𝑥 − 3
≤
2
𝑥
𝑓(𝑥) =
𝑥2 + 2𝑥 + 3
𝑥 + 1
A rational equation or inequality can be solved for all 𝑥 values that satisfy the
equation or inequality. Whereas we solve an equation or inequality, we do not “solve"
functions. Rather, a function (and in particular, a rational function) expresses a

21
relationship between two variables (such as 𝑥 and 𝑦), and can be represented by a
table of values or a graph.
Activity 3. Determine whether the given is a rational function, rational equation,
rational inequality, or none of these.
1. 𝑦 = 5𝑥3
− 2𝑥 + 1 6. 6𝑥 −
5
𝑥+3
≥ 0
2.
8
𝑥
− 8 =
𝑥
2𝑥−1
3. √𝑥 − 2 = 4
4.
𝑥−1
𝑥+1
= 𝑥3
5. 𝑔(𝑥) =
7𝑥3−4√𝑥+1
𝑥2+3
Activity 4. Find the solutions for each rational equation below. Make sure to check
for extraneous solutions.
1.
3
𝑥+1
=
2
𝑥−3
4.
𝑥2−4𝑥
𝑥−2
=
14−9𝑥
𝑥−2
2.
2𝑥
𝑥+1
+
5
2𝑥
= 2 5.
2𝑥−1
𝑥+3
= 5
3.
𝑥2−10
𝑥−1
=
−14−5𝑥
𝑥−1
Activity 5. Find the solution set for each rational inequality below. Graph the
solution set on a number line.
1.
(𝑥+3)(𝑥−2)
(𝑥+2)(𝑥−1)
≥ 0 4.
𝑥−2
𝑥2−3𝑥−10
< 0
2.
(𝑥+4)(𝑥−3)
(𝑥−2)(𝑥2+2)
≥ 0 5.
𝑥−1
𝑥+3
> 0
3.
𝑥+1
𝑥+3
≤ 2 6.
𝑥2−𝑥−30
𝑥−1
≥ 0
Activity 6. Sketch the graph of the following function.
1. 𝑓(𝑥) −
9
𝑥2−9
2. 𝑓(𝑥) =
𝑥2−4
𝑥2−4𝑥

22
What I Have Learned
1. An ________________ of a rational function is a point where the graph of the
rational function intersects the x- or y-axis.
2. Zeroes are also known as____________________, solutions or roots of functions.
3. The vertical line x = a is a vertical asymptote of a function f if the graph of f either
__________________________ without bound as the x-values approach a from the
right or left.
4. The horizontal line ________________is a horizontal asymptote of the function f if
f(x) gets closer to b as x increases or decreases without bound ( 

x ).
5. There are three possibilities in determining horizontal asymptote(s) of a rational
function.
 If n < m, the horizontal asymptote is_____________________.
 If n = m, the horizontal asymptote is ____________________ , where a is the
leading coefficient of the numerator and b is the leading coefficient of the
denominator.
 If n > m, _____________________ horizontal asymptote.
6. A _____________________ is an expression that can be written as a ratio of
two polynomials.
7. A ___________________ is an inequality involving rational expressions.
8. ______________________ is a function of the form 𝑓(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
where 𝑝(𝑥)and
𝑞(𝑥) are polynomial functions and 𝑞(𝑥) is a nonzero polynomial.
9. A ___________________ is an equation containing at least one ratio of two
polynomials.
10.To solve a rational equation, eliminate denominators by multiplying each
term of the equation by the ___________________________.
What I Can Do
Rational functions abound in real life, we just don't always think of them
that way. Read and understand the problem carefully.
Suppose you are buying face mask for yourself, your friends, and family
during this Covid-19 pandemic. The face mask shop has a deal going, if you buy
one facemask for 35 pesos, then additional face masks are only 30 pesos each. As
you buy more and more face masks (more and more as the health risk arise!), what
is the average cost per face mask?

23
Represent this situation into a rational equation showing the price per face
mask based on number purchased, and the number of face masks. Determine its
horizontal asymptotes and explain what this represents.
Assessment
Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a
separate sheet of paper.
1. Point/s where the graph of the rational function intersects the x- or y-axis.
a. range c. intercepts
2. The function value when x=0.
3. The values of x which make the function zero.
a. range c. zeroes
4. The real numbered zeroes are also _____________ of the graph of the function.
5. If the graph of function either increases or decreases without bound as the x-
values approach a from the right or left, the line is called
a. vertical asymptote c. x - axis
b. horizontal asymptote d. y – axis

24
6. The function
8
4
)
( 2
2


x
x
x
f has no vertical asymptote because,
a. vertical asymptotes will occur at those values of x for which the
denominator is equal to zero.
b. 8
 is not a real number
c. 8 is not a perfect square
d. a and b
7. Setting )
(x
q =0 is the primary step in determining,
a. domain c. horizontal asymptote
b. vertical asymptote d. intercept
8. If the degrees of the numerator (n) is less than the degrees of the
denominator (m), y = 0 is
a. vertical asymptote c. zero of the function
b. horizontal asymptote d. y – intercept
9. Let n be the degree of the numerator and m be the degree of the denominator.
If n > m,
a. there is no horizontal asymptote c. horizontal asymptote is
b
a
y 
b. horizontal asymptote is y = 0 d. a and b
10.The zero/es of the function
)
2
)(
1
(
)
1
)(
1
(
)
(





x
x
x
x
x
f is/are
a. x = -1, -2 c. x = -2
b. x = 1, 2 d. x = 1
For numbers 11-14, consider the function
1
4
)
( 2
2



x
x
x
f with its graph below.
11.The x- and y-intercept is
a. (2,0), (-2,0) and (0, 4)
b. 2, -2 and 4
c. 2 and 4
d. (2,0), (-2,0)
12.The vertical asymptotes are
a. -1 and 1
b. -2 and 2
c. 2 and 4
d. -2 and 1

25
13.What is the horizontal asymptote of the given function?
a. y = -1 c. y = 2
b. y = 1 d. y = -2
14.What are the zero/es of given function?
a. y = - 2, 2 c. x = (2,0), and (-2,0)
b. x = 2, -2 d. b and c
15. On what interval(s) is
𝑥+10
3𝑥−2
≤ 3?
𝑎. (−∞,
2
3
) 𝑎𝑛𝑑[1, ∞) c. (−∞,
2
3
) 𝑜𝑟 [2, ∞)
𝑏. (−∞,
2
5
) 𝑜𝑟 [2, ∞) d. (−∞,
2
3
) 𝑎𝑛𝑑[2, ∞)
Additional Activities
This section includes supplementary activities related to the intercepts, zeroes and
asymptotes of inverse of rational functions.
This section includes supplementary activities related to rational functions,
equations and inequalities.
1. Give examples of problems or situations in real life that involve the use
of rational equation, inequality, and function. In each example,
a. explain the problem or situation.
b. solve the problem.
discuss how you can use these sample situations in your daily life,
especially in formulating conclusions and/or making decisions
2. Show that
7
5
)
( 2
2


x
x
x
f has no vertical asymptote algebraically.
Determine its intercepts, zeroes and horizontal asymptote.

26
Answer Key
What’s
More
Activity
3.
1.
Rational
Function
2.
Rational
Equation
3.
None
of
these
4.
Rational
Equation
5.
None
of
these
6.
Rational
Inequality
Activity
4.
1.
11
2.
-5
3.
-4,
-1
4.
-7
5.
−
16
3
Activity
5.
1.
(
−∞,
3
∪
[2,
∞)
2.
[
−4,2
)
∪
[3,
∞)
3.
(−∞,
−5
∪
(−3,
∞)
4.
(−∞,
−2)
∪
(2,5)
5.
(
−∞,
−3
)
∪
[1,
∞)
6.
[
−5,
−1
)
∪
[
6,
∞
)
Activity
6.
1.
𝑦-intercept:(0,-1)
no
𝑥−intercept
asymptotes:𝑥
=
±3
and
𝑦
=
0
2.
no
𝑦−intercept
𝑥-intercept:
𝑥
=
±2
asymptotes:
𝑥
=
0,
𝑥
=
4and
𝑦
=
1
What's
More
Activity
1.
COMPLETE
ME
1.

(-5,
0)
or
x=-5

(0,-1)

x=5

y=1
2.

(-0.5,
0)
or
x=-5

B(0,0.5)

x=-1,
x=-2

y=4
3.

(-1.333,0)
or
x=-4/3

(0,4)

x=-1,
x=-0.5

y=0
Activity
2.
LOCATE
ME
A.

(-2,0)

(0,1)

(-2,0)
or
x=-2

x=2

y=1
B.

(-4,0)
and
(4,0)

(0,4)

(-4,0)&(4,0)
or
x=-4,4

x=-2,2

y=1
What
I
Know
1.
C
2.
D
3.
B
4.
C
5.
A
6.
A
7.
C
8.
A
9.
B
10.C
11.A
12.B
13.C
14.D
15.A

27
What
Have
I
Learned
1.
intercepts
2.
x-intercepts
3.
increasing
or
decreasing
4.
y=b
5.

y=0

y=a/b

there
is
no
6.
rational
expression
7.
rational
inequality
8.
rational
function
9.
rational
equation
10.
LCD
Assessment
1.
C
2.
B
3.
C
4.
A
5.
A
6.
B
7.
B
8.
B
9.
A
10.C
11.A
12.A
13.B
14.D
15.D

28
References
Department of Education – Bureau of Learning Resources (DepEd-BLR) (2016)
General Mathematics Learner’s Material. Lexicon Press Inc., Philippines
Department of Education – Bureau of Learning Resources (DepEd-BLR) (2016)
General Mathematics Teacher’s Material. Lexicon Press Inc., Philippines
Graphing Calculator. Desmos. Accessed June 11, 2020.
https://www.desmos.com/calculator
Learning, Lumen. “College Algebra.” Lumen. Accessed June 11, 2020.
https://courses.lumenlearning.com/waymakercollegealgebra/chapter/doma
in-and-vertical-asymptotes/#:~:text=A vertical asymptote represents a,the
denominator to equal zero.
Math is Fun Advanced. Solving Rational Inequalities. Accessed on June 11, 2020.
https://www.mathsisfun.com/algebra/inequality-rational-solving.html
Precalculus. Rational Functions by Jay Abramson, et al. Accessed on June 11, 2020
https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/solve-
applied-problems-involving-rational-functions/
Rational Functions Problems. Accessed on June 11, 2020.
https://www.onlinemathlearning.com/rational-function-problems.html

29
DISCLAIMER
This Self-learning Module (SLM) was developed by DepEd SOCCSKSARGEN with
the primary objective of preparing for and addressing the new normal. Contents
of this module were based on DepEd’s Most Essential Learning Competencies
(MELC). This is a supplementary material to be used by all learners of Region XII
in all public schools beginning SY 2020-2021. The process of LR development
was observed in the production of this module. This is version 1.0. We highly
encourage feedback, comments, and recommendations.
For inquiries or feedback, please write or call:
Department of Education – SOCCSKSARGEN
Learning Resource Management System (LRMS)
Regional Center, Brgy. Carpenter Hill, City of Koronadal
Telefax No.: (083) 2288825/ (083) 2281893
Email Address: region12@deped.gov.ph

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