Genmath Solving Rational Functions genera math

Lesson 4
Solving Problems
Involving Rational
Functions

Objectives
At the end of this lesson, the learner should be able to
● accurately write a rational function to represent a
real-life situation; and
● accurately find the value of a rational function at a
given value of .

Essential Questions
● How will rational functions be used to model a real-life
situation?
● How will the domain and range of a rational function be
used in order to solve real-life problems?
● How will rational functions be evaluated at specified
values of the variable?

Warm Up!
For this lesson, we will discuss solving problems involving
rational functions. Before we proceed, let us watch this video
to investigate about fixed and variable costs.
(Click on the link below to watch the video.)
Glamour. “How a 24-Year-Old Making $75K in NYC Spends Her
Money | Money Tours | Glamour”. Retrieved 10 April 2019
from https://www.youtube.com/watch?v=Hnrfv4nR-DM

How do you spend your money?
How much can you control your expenses?

Guide Questions
● Do you think the woman in the video spends her money
wisely? Which areas can she make adjustments to if she
wants to save money?
● What is the difference between fixed and variable costs?
● How can we use rational functions to represent variable
costs?

Learn about It!
Rational function
a function of the form , where and are both polynomials and
1
Example:
The function is a rational function since both the numerator
and denominator are polynomials, and the denominator is
not equal to zero.

Learn about It!
Asymptote
a line that a curve approaches, but does not intersect
2
Example:
Consider the rational function .
The vertical asymptote of is . If , the denominator will
become 0. Thus, the graph will never intersect the line .

Learn about It!
Asymptote
2
Example:
Consider the rational function .
Meanwhile, the horizontal asymptote of is the line
. As becomes larger, the value of smaller, eventually
approaching .

Learn about It!
Domain of a function
the set of all values of that have corresponding values of ; it contains all values
that go into the function
3
Example:
The domain of the rational function is the set of all real
numbers such that . In symbols, .

Learn about It!
Example:
The range of the rational function is the set of all real
numbers such that . In symbols, .
Range of a function
the set of all values of that can be obtained from the possible values of ; it
contains all possible values of the function
4

Try It!
Example 1: A parking area has a fixed entrance fee of ₱50,
plus ₱25 for each hour of stay. Represent the average cost
per hour using a rational function.

Try It!
Solution: The average cost can be calculated by finding the
sum of the fixed entrance fee and the variable cost, then
dividing the sum by the number of hours of stay.
If the fixed entrance fee is ₱50 and the variable cost is ₱25,
then total cost after hours is .

Try It!
Solution: The total cost after hours is . Dividing it by the
total number of hours gives us the average cost per hour.
Thus, the function of average cost per hour is given by:

Try It!
Example 2: A movie rental store charges an initial fee of
₱250, then ₱50 for each movie you rent. What is the average
cost function and the average cost per movie if you rent 5
movies?

Try It!
movies?
Solution: The average cost function can be expressed as the
sum of the initial fee and the variable cost, divided by the
number of movies rented. Let be the number of movies
rented.

Try It!
movies?
Solution: Let be the number of movies rented.
If the initial fee is ₱250 and the variable cost is ₱50, then the
total cost after movies is

Try It!
movies?
Solution: The total cost after movies is
Dividing it by the total number of movies gives us the
average cost per movie.

Try It!
movies?
Solution: Thus, the function of average cost per movie is
given by:

Try It!
movies?
Solution: To determine the average cost per movie if you
rent 5 movies, substitute into the average cost function.

Try It!
movies?
Solution:

Try It!
movies?
Solution:
Therefore, the average cost for each of five movies is ₱100.

Let’s Practice!
Individual Practice:
1. Christy charges a fixed registration fee of ₱150 for a piano
lesson, plus ₱80 per session. Represent the average cost
per session using a rational function.
2. A beach resort charges ₱3 500 upon check-in, plus ₱800
daily. What is the average cost function and the average
daily cost if you stay for 5 days?

Let’s Practice!
Group Practice: To be done in 2 to 5 groups
Rosie receives a fixed daily salary of ₱650 as a teaching
assistant at a private high school. She also receives ₱50 for
each hour of work. Represent the average salary per hour
using a rational function. If Rosie works for 5 hours, what
will be her average salary per hour?

Key Points
Rational function
a function of the form , where and are both polynomials and
1
Asymptote
2

Key Points
Domain of a function
the set of all values of that have corresponding values of ; it contains all values
that go into the function
3
Range of a function
the set of all values of that can be obtained from the possible values of ; it
contains all possible values of the function
4

Synthesis
● How can you describe and represent a situation using
rational functions?
● What are the difficulties you encounter while solving
problems involving rational functions? How did you
overcome them?
● How to we check if a given value of a variable is a solution
to a rational equation?

Genmath Solving Rational Functions genera math

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