RATIONAL FUNCTIONS SHS 11

1. GENERAL MATHEMATICS
Rational Functions

2. Test I: Choose the letter of the correct answer and write it on a separate sheet of paper.
1. Martin can finish a job in 6 hours working alone. Victoria has more experience
and can finish the same job in 4 hours working alone. How long will it take both
people to finish that job working together?
A. 2.4 hours
B. 2.9 hours
C. 3.5 hours
D. 3.7 hours
2. Sarah can finish a job in 7 hours working alone. If Sarah and Matteo work
together, they can finish the work in 3 hours. How long will it take if Matteo will
choose to work alone?
A. 4.25 hours
B. 4.85 hours
C. 5.25 hours
D. 5.55 hours

8. REAL LIFE FUNCTIONS
L E S S O N 5

9. •How do we find the Least Common
Denominator (LCD)?
•To add or subtract fractions with
different denominators, you must find
the least common denominator. LCD
refers to the lowest multiple shared by
each original denominator in the
equation, or the smallest whole
number that can be divided by each
denominator.

11. How to solve problems involving Rational
Functions?
Example 1:
Martin can finish a job in 6 hours working alone.
Victor has more experience and can finish the
same job in 4 hours working alone. How long will it
take both people to finish that job working together?

12. Given:
6 hours – Martin can do the work alone
4 hours – Victor can do the work alone
Find: x – hours Martin and Victor can do
the work
Solution:

13. •Therefore, it will
take 2.4 hours for
Martin and Victor
to do the work
together.

14. Example 2:
Sarah can finish a job in 7 hours working alone.
If Sarah and Matteo work together, they can
finish the work in 3 hours. How long will it take if
Matteo will do the work alone?
Given:
7 hours – Sarah can do the work alone
3 hours – Sarah and Matteo can do the work
together
Find: x hours – for Matteo to do the work alone

15. Solution:

17. Directions: Solve each problem. Show your solutions
and write answers on a separate sheet of paper.
1. One person can complete a task in 8
hours. Another person can complete a task
in 3 hours. How many hours does it take for
them to complete the task if they work
together?
2. Sigfried can paint a house in 5 hours.
Stephanie can do it in 4 hours. How long will
NOW IT’S YOUR TURN!

18. 3. Joy can pile 100 boxes of goods in 5 hours.
Stephen and Joy can pile 100 boxes in 2
hours. If Stephen chooses to work alone,
how long will it take?
4. Computer A can finish a calculation in 20
minutes. If Computer A and Computer B can
finish the calculation in 8 minutes, how long
does it take for the Computer B to finish the
calculation alone?

19. 5. Aiah can paint a house in 3 hours.
Princess can do it in 4 hours. How
long will it take the two working
together?
4. Carlo can wash a load of dishes in
35 minutes. If JL works together with
Carlo it takes them 20 minutes to
wash the same load of dishes. How
long will it take JL to wash a load of
dishes by himself?

20. Remember…

21. RATIONAL
FUNCTIONS,
EQUATIONS AND
INEQUALITIES
•Lesson 6

22. The table below shows the definitions of rational functions, rational equations and rational
inequalities with examples.
DEFINITION OF TERMS

23. NOW IT’S YOUR TURN!
Directions: Identify whether the following is a rational
function, rational equation or rational inequality.

24. •Rational Function is a function of the form of
𝑓( )= ( )/ ( ) where p(x) and q(x) are polynomials,
𝑥 𝑝 𝑥 𝑞 𝑥
and q(x) is not the zero function.
• Rational Equation is an equation involving rational
expressions.
• Rational Inequality is an inequality involving rational
expressions.
Remember…

25. RATIONAL
EXPRESSIONS AND
EQUATIONS
G E N E R A L M AT H E M AT I C S

26. Lesson Objectives
At the end of the lesson, the students must be
able to:
• distinguish rational function, rational
equation, and rational inequality;
• represent real-life situations using rational
functions; and
• solve rational equations and inequalities.

27. RATIONAL EXPRESSION
A rational expression can be written in the form
where A and B are polynomials and B ≠ 0.
Examples

28. Simplifying Rational Expression
1. Factor the numerator and denominator.
2. Write a product of two rational expressions,
one factor containing the GCF of the
numerator and denominator, and the other
containing the remaining factors.
3. Rewrite the factor containing the GCF as 1.
4. Multiply the remaining factors by 1.

29. EXAMPLE 1
Simplify
Solution

30. ADDING AND SUBTRACTING RATIONAL EXPRESSION
WITH LIKE DENOMINATORS
1. Add (or subtract) the numerators.
2. Retain the common denominator.
3. Simplify the result
Adding and Subtracting Rational
Expression with Different Denominators
1. Find the least common denominator (LCD).
2. Write the equivalent expression of each rational expression.
3. Add or subtract the numerators and keep the LCD.
4. Simplify the result, if possible..

31. EXAMPLE 2A
Find the sum of and .
Solution
b
a
4
2

32. EXAMPLE 2B
Add and .
Solution

33. MULTIPLYING RATIONAL EXPRESSION
1. Multiply the numerators and denominators.
2. Write the resulting numerator and denominator in
factored form.
3. Divide out any numerator factor with matching
denominator factor.
4. Simplify, if possible.

34. EXAMPLE 3
Multiply and .
Solution

35. RATIONAL EQUATION
A rational equation is an equation
that contains one or more rational
expressions.
Example

36. SOLVING RATIONAL EQUATION
1. Eliminate the rational expressions in the
equation by multiplying both sides of the
equation by the LCD.
2. Solve the equation.
3. Check your solution.

37. EXAMPLE 4A
Solve for x.
Solution

38. EXAMPLE 4B
Solve for x.
Solution

39. EXERCISES
Solve for x.
1. 4.
2. 5.
3.
6.
3
1
5
2
9


y

40. EXERCISES
Solve for x.
1.
2.
3.
3
5
5
1



x
x

41. NOW IT’S YOUR TURN!
Solve the following rational equations and
inequality. Show your solutions and write your
answers on a separate sheet of paper.

42. •What is the
difference between
Rational Equation
and Inequalities?

43. • A rational equation is an equation that
contains one or more rational
expressions while a rational inequality
is an inequality that contains one or
more rational expressions with
inequality symbols ≤, ≥, <, >, and ≠.

44. REPRESENTATIONS OF
RATIONAL FUNCTIONS
•Lesson 8

45. REVIEW
These are the terms or group of terms you need to know
before going to the discussion on the domain and range of
rational functions.
1. Set of Real Numbers ( ) – The real numbers include
ℝ
natural numbers or counting numbers, whole numbers,
integers, rational numbers (fractions and repeating or
terminating decimals), and irrational numbers. The set of real
numbers consists of all the numbers that have a location on
the number line.
2. Domain – the set of all x – values in a relation.
3. Range – the set of all y – values in a relation.

46. •4. Degree of Polynomial – the
degree of a polynomial with one
variable is based on the highest
exponent.
•For example, in the expression x3 +
2x + 1, the degree is 3 since the
highest exponent is 3.

48. •The domain of the
rational function is
𝑓
the set of real numbers
except those values
of x that will make the
denominator zero.

49. How to find the Domain of a Rational Function? The
domain of a function consists of the set of all real number
( ) except the value(s) that make the denominator zero.
ℝ

50. How to find the Domain of a Rational Function? The
domain of a function consists of the set of all real number
( ) except the value(s) that make the denominator zero.
ℝ

51. How to find the Range of a Rational Function? The range of a rational
function f is the set of real numbers except those values that fall to the
following conditions.

52. How to find the Range of a Rational Function? The range of a rational
function f is the set of real numbers except those values that fall to the
following conditions.

54. REMEMBER

55. NOW IT’S YOUR TURN!

57. Lesson
9
Zeroes and
y-Intercepts of
Rational
Functions

58. At the end of this lesson, you are expected to:
Represent real life situations using rational
functions; and
determine the zeroes and y-intercepts of
rational functions.

59. Let’s start with a
review about
rational function.

61. What are ZEROES
and y-INTERCEPTS
of a Function?

62. Zeroes:
These are the values of x which make the function zero.
The real numbered zeroes are also x-intercepts of the
graph of the function.
y- Intercept:
The y-intercept is the function value when the value of x is
equal to 0.
(Versoza et al., General Mathematics Teaching Guide 2016)

64. Solution: (a) zeroes
Since the zeroes or x-intercepts of a rational function are the
values of x that will make the function zero. A rational
function will be zero if its numerator is zero. Therefore, the
zeroes of a rational function are the zeroes of its numerator.
Step 1: Take the numerator and equate it to zero. x - 3 = 0
Step 2: Find the value of x using Addition Property of
Equality, add +3 to both sides. x – 3 + 3 = 0 + 3 , simplify x =
3 Therefore, 3 is a zero of f(x).

67. (a) zeroes To find the zeroes of this function, let’s equate the
numerator to zero. The zeroes of a rational function are the
zeroes of its numerator.
Step 1: Take the numerator (−3 +2) and equate it to zero.
𝑥
−3 +2=0
𝑥
Step 2: Find the value of x by factoring (x-2)(x-1) =0
Step 3: Equate each factor to zero and solve for x.

70. NOW IT’S YOUR TURN!

72. Remember
• Zeroes These are the
values of x which
make the function
zero.
• y- Intercept The y-
intercept is the
function value when
the value of x is equal
to 0 (x=0).

73. Lesson 11
Asymptotes
of Rational
Functions

74. At the end of this lesson, you are expected
to:
o differentiate horizontal and vertical
asymptotes; and
o determine the horizontal and vertical
asymptotes of rational functions.

75. Asymptotes of Rational
Functions: Consider the
graph shown. Observe
that the graph approaches
closer and closer but
never touches an
imaginary line. Can you
spot where that imaginary
line is? Draw a broken
line to locate that line. I
know you can do it!

76. What is an
ASYMPTOTE?

77. •An asymptote is a line that a curve approaches or
gets closer and closer to but never touches. It could
be a horizontal, vertical, or slanted line (Math is Fun
2020).

78. Vertical
Asymptotes
Definition:
The vertical line x = a is a
vertical asymptote of a function
f if the graph of f either
increases or decreases without
a bound as the x-values
approach a from the right or
left (Versoza et al., General
Mathematics Teaching Guide
2016).

79. Finding the
Vertical
Asymptote
s of a
Rational
Function
•(a) Reduce the rational function
to lowest terms by cancelling out
the
•common factor/s in the
numerator and denominator.
•(b) Find the values a that will
make the denominator of the
reduced rational function equal
to zero.
•(c) The line x = a is the vertical
asymptote.

83. Graphically, the
vertical
asymptote is
the broken line
as shown
below:

84. Horizontal
Asymptotes
•Definition:
The horizontal line y = b is a
horizontal asymptote of a function
f if f(x) gets closer to b as x
increases or decreases without
bound ( + )
𝑥 → ∞ 𝑜𝑟 𝑥 → −∞
(Versoza et al., General
Mathematics Teaching Guide 2016).

85. •*A rational function may or may not
cross its horizontal asymptote. If the
function does not cross the horizontal
asymptote y=b, then b is not part of
the range of the rational function.

88. •Graphically, the horizontal asymptote is the broken line
as shown below:
•Example 4:

90. NOW IT’S YOUR TURN!
SODOKU: ASYMPTOTES Instructions: Copy the sodoku puzzle and the table below on a separate
sheet of paper. First, fill in the needed information to complete the table. When done, go back to
the puzzle to solve it. Remember that each column, row and 3x3 mini squares should consist of
numbers 1-9 with no repetition. Only the positive integer solution is used in the puzzle and indicate
the sign in the table below.

92. Remember
•• Vertical Asymptote
•o The vertical line x = a is a vertical
asymptote of a function f if the
graph of f either increases or
decreases without a bound as the x-
values approach a from the right or
left.
•• Horizontal Asymptote
•o The horizontal line y = b is a
horizontal asymptote of a function f
if f(x) gets closer to b as x increases
or decreases without bound (𝑥 →
+ )
∞ 𝑜𝑟 𝑥 → −∞

93. Lesson 12
Solving Problems Involving Rational
Functions, Equations and Inequalities

94. At the end of this lesson,
you are expected to:
o solve problems involving
Rational Functions,
Equations and Inequalities

96. Let’s solve this problem.
A mobile phone user is charged P300 monthly
for a particular plan, which includes 100 free text
messages. Messages in excess of 100 are charged
P1 each. a) Represent the monthly cost for text
messaging using the function t(m), where m is the
number of messages sent in a month. b) How much
is the charge if the user sent 130 text messages?
Monthly Cost: ______________ (b) Charge: ______________

97. PROBLEM SOLVING
Example 1: You want to join an
online calligraphy class. You will
pay an annual membership fee
of Php 500.00, then Php 150.00
for each class you go to. What
is the average cost per class if
you go to 10 classes?

98. Step 1: Understand the Problem. Given:
Php 500.00 - annual membership fee Php
150.00 - payment per class you attend 10
classes - number of classes Find: What is
the average cost per class if you go to 10
classes?

106. To get the table of signs, just
substitute the values of the test point to
the numerator, denominator, and the
whole rational function. The sign of the
answer will be written in the table of
signs as shown in the in the previous
From (a) to (c) and table of signs,

107. we can now sketch the graph like the
one shown below:
(e) Based on the graph, the range of the
function is {y | ≠1}
𝜖𝑅 𝑦

108. NOW IT’S YOUR TURN!

109. •2. Jonathan’s badminton team must collect
at least 160 shuttlecocks for their practices
in preparation for Palarong Pambansa. The
team members brought 42 shuttlecocks on
Monday and 65 shuttle cocks on
Wednesday. How many more shuttlecocks
must the team bring to meet their goal?

110. Remember
• In solving problems, follow the steps to
have a systematic way of answering.
• In sketching a rational inequality, identify
the (a) domain, (b) intercepts, (c) asymptotes
and (d) table of signs of the given inequality.

111. Assessment (Post-test)

112. Test I. Choose the letter of the correct answer and
write them on a separate sheet of paper.
1. Melvin can finish a job in 9 hours working alone.
Vanessa has more experience and can finish the same
job in 6 hours working alone. How long will it take both
people to finish that job working together?
A. 2.3 hours
B. 2.9 hours
C. 3.6 hours
D. 3.9 hours
2. Liza can finish a job in 5 hours working alone. If Liza
and Enrique work together, they can finish the work in
3 hours. How long will it take if Enrique will choose to
work alone?
A. 10 hours
B. 7.5 hours
C. 7 hours
D. 6.5 hours