This document discusses rational functions and their representations. It begins with an introduction to rational functions and examples. It then provides practice problems representing real-world situations with rational functions. The document discusses distinguishing between rational functions, equations, and inequalities. It provides examples and practice problems for students to work through. The learning objectives are outlined which include defining and working with rational functions and expressions.

1. What’s Your Idea?
What word/s do you think of when you hear
the word “rational functions”?

2. Let’s
Analyze!

3. Answer This!
▪How much is the total amount of money the
barangay can use for its relief operations?
▪What is the concern of one the Barangay
Kagawad regarding the total number of
families who will benefit with the relief? Do
you think it is valid? Why?

4. Answer This!
▪Create a model or equation that will represent
to the amount of relief each family will receive
bearing in mind that the number of families
will vary.
▪After resolving the amount of relief each
family may receive, what other problem may
arise in the current situation?

5. Answer This!
▪If you are one of the residents of that
barangay what will be your participation to
help the officials?

6. Session Objectives
At the end of the session, the students should
be able to:
1. Define rational function
2. Recall the concepts of polynomial function
3. Represent real-life situations using rational
functions

7. Session Objectives
4. Describe a rational expression
5. Distinguish whether an expression is
rational or not
6. Distinguish rational function, rational
equation and rational inequality

8. Rational Functions
Representing Real-life Situations Using
Functions

9. Remember This!
▪ The following are
examples of rational
functions:
1.𝑟 𝑥 =
𝑥3−1
𝑥+1
, 𝑥
2.𝑓 𝑥 =
1
𝑥
, 𝑥 ≠ 0

10. Try This!
Directions: Try to represent the given examples with rational
functions.
1.The Local Government Unit allotted a budget of ₱100,000.00
the feeding program in the Day Care Center. The amount will
divided equally to all the pupils in the Day Care Center. Write
equation showing the relationship of the allotted amount per
pupil represented by f(x) versus the total number of children
represented by x.

11. Try This!
2. Suppose a benefactor wants to supplement
the budget allotted for each child by
donating additional ₱650.00 per child. If
h(x) represents the new amount allotted per
child, construct a function representing the
relationship.

12. Try This!
3. A car is to travel a distance of 70
kilometers. Express the velocity (v) as a
function of travel time (t) in hours.

13. Rational Functions
Rational Functions, Rational
Equations, and Rational Inequalities

14. Know the Difference

15. Try This!
Directions: Determine whether the given sentence is
a rational equation, a rational function, a rational
inequality or none of these.
1.
𝑥+5
𝑥−1
= 𝑦
2.
2
𝑥+1
≤ 3

16. Learning Task 1
Create three examples each for rational
function, rational equation, and rational
inequality.

17. Learning Task 2
Read and analyze the situation below then answer the question
given.
In an inter-barangay basketball league, the team from Hermana
Fausta has won 12 out of 25 games, a winning percentage of 48%.
We have seen that they need to win 8 games consecutively to raise
their percentage to 60%. What will be their winning percentage if
they win 10, 20, 30, 50, 100 games? Will they reach a 100%
winning percentage? Make sure to create an appropriate model /
equation for the given problem.

18. Home-based Learning Task 1
You conducted an outreach activity to help the
needy in your community and have solicited
₱52,000.00. You wanted to propose a plan on how
to equally divide the money and the possible relief
goods that will be included. If you will make a
proposal what plan will you do? Show your plan by
filling up the form below:

19. Home-based Learning Task 1

20. Home-based Learning Task 1

21. Session Objectives
At the end of the session, the students should be able
to:
1. Apply appropriate methods in solving rational
equations and inequalities
2. Solve rational equations and inequalities using
algebraic techniques for simplifying and
manipulating of expressions
3. Determine whether the solutions found are
acceptable for the problem by checking the
solutions

22. Session Objectives
4. Construct the equation, table of values and
graph of a rational function
5. Represent a rational function using
equation, table of values and graphs
6. Define domain and range
7. Find the domain and range of a rational
function

23. Rational Functions
Solving Rational Equations and
Rational Inequalities

24. Solving Rational Equations

25. Solving Rational Equations

26. Solving Rational Inequalities

27. Solving Rational Inequalities

28. Solving Rational Inequalities

29. Rational Functions
Representations of
Rational Functions

30. Let’s Try This!
1. Pueblo Por La Playa is a 12.5 hectare Mexican-
inspired exclusive leisure club nestled off the calm,
clear waters of Pagbilao Quezon. The "Pueblo"
offers the total leisure and recreation experience for
the entire family. Since it is an exclusive resort, it
has a membership fee. Pueblo Por La Playa charges
a ₱300,000.00 annual fee, then ₱700.00 for each
day you stay there. Find the average cost per day to
stay in the resort in 5, 10, 15 and up to 30 days.
Graph the function to show whether it forms a
straight line or a curve.

31. Let’s Try This!
a. Define a formula for the average cost for every
5 days to stay in the resort f(x).
Hint: Since the problem asks for the average cost,
use the formula in getting an average.
b. Based from the situation above, complete the
following table to show the average cost every 5
days.

32. Let’s Try This!
x 0 5 10 15 20 25 30
y 0
Hint: Substitute the value of x in your
equation
c. Plot the following points on the cartesian
plane
To graph, simply plot the points and connect
it by a smooth curve line.

33. Rational Functions
Domain and Range of
Rational Functions

34. Remember This!
▪In definition, the domain of a function is
the set of all values that the variable x
can take while the range of a function is
the set of all values that y or f(x) can
take.

35. Remember This!
▪We can write the domain and range
using different forms:
▪By roster format - this method
enumerates the lists of all values in
the set. Ex. The domain of r(t) are (1,
1.25, 1.5, 1.75, 2).

36. Remember This!
▪ By set-builder form or notation - for example, in
numbers 10 to 20. you can say {x | x are even
numbers from 10 to 20). The | is read as “such that.”
Assuming that you also include odd numbers in the
domain from 10 to 20, then, you can write the
domain of the function D(x) as {x | x ϵ R, 10≤x≤20},
read as “x such that x is an element of a real number
wherein x is greater than or equal to 10 but less than
or equal to 20.”

37. Remember This!
▪ By interval notation – for example, in a function
f(x) = 5/(x-3), the domain of this function can be
written in the form, (-∞, 3) U (3, ∞). This means
that the values of the domain can take all real
values of x except 3, otherwise the function is
undefined.

38. Let’sTryThis!
Directions: Find the domain and range of the following
rational functions.
1.f(x) =
2𝑥−3
𝑥2
2.f(x) =
𝑥−2
𝑥+2
3.f(x) =
𝑥2−3𝑥−4
𝑥+1