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STUDY GUIDE

GENERAL MATHEMATICS | UNIT 4

Rational Functions

Table of Contents

Introduction ...................................................................................................................................... 3

Test Your Prerequisite Skills .......................................................................................................... 4

Objectives ........................................................................................................................................ 5

Lesson 1: Introduction to Rational Functions

- Warm Up! ............................................................................................................................. 5

- Learn about It! ..................................................................................................................... 6

- Let’s Practice! ....................................................................................................................... 7

- Check Your Understanding! ............................................................................................. 10

Lesson 2: Representing Rational Functions through Tables, Graphs, and Equations

- Warm Up! ........................................................................................................................... 12

- Learn about It! ................................................................................................................... 13

- Let’s Practice! ..................................................................................................................... 16


Lesson 3: Domain and Range of a Rational Function

- Warm Up! ........................................................................................................................... 23

- Learn about It! ................................................................................................................... 24

- Let’s Practice! ..................................................................................................................... 27


Lesson 4: Solving Problems Involving Rational Functions

- Warm Up! ........................................................................................................................... 35

- Learn about It! ................................................................................................................... 35


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STUDY GUIDE

- Let’s Practice! ..................................................................................................................... 37


Challenge Yourself! ....................................................................................................................... 41

Performance Task ......................................................................................................................... 42

Wrap-up ......................................................................................................................................... 44

Key to Let’s Practice! ...................................................................................................................... 45

References ..................................................................................................................................... 47


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STUDY GUIDE

GRADE 11| GENERAL MATHEMATICS

UNIT 4

Rational Functions You have been studying functions in the previous lessons. One kind of functions that are widely used to model and solve many problems in the business world is the rational functions. Some examples of real-world scenarios that involve rational functions include average speed, concentrations of mixtures, average sales over time, and average costs over time. Many practical situations can be represented and analyzed using mathematical equations. For example, a person's Body Mass Index or BMI can give us an idea of the state of his or her health. This can be calculated by dividing the person's weight in kilograms by the square of his or her height in meters.

Another quantity that can be represented using rational functions is the expenses of a factory for a product. For example, a bag company needs to spend ₱20 000 monthly to start their business operation. For every bag they produce, they need to

spend another ₱400. How much do they spend to produce a bag? How much will they spend in all if they need to make 200 bags a day for a month? The average cost 𝐶(𝑥) can be calculated by finding the sum of the fixed monthly cost and the variable cost, then dividing the sum by the number of bags produced. In this unit, we will learn about rational functions and its application to real-life situations.

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STUDY GUIDE

Before you get started, answer the following items on a separate sheet of paper. This will help you assess your prior knowledge and practice some skills that you will need in studying the lessons in this unit. Show your complete solution.

1. Identify the following expressions are polynomial or not.

a. 3𝑥& + 2 b. −*

&𝑥& − 𝑥 + √5

c. 𝑥 + &-

d. 2- − 1 e. 4𝑥& − 𝜋𝑥 − 10 f. 7𝑥3 − √𝑥 + 7

2. Graph the following polynomial

functions. a. 𝑓(𝑥) = 3𝑥 + 5 b. 𝑓(𝑥) = 4𝑥 − 1 c. 𝑓(𝑥) = 𝑥& + 2𝑥 − 5 d. 𝑓(𝑥) = 3𝑥& + 𝑥 − 2 e. 𝑓(𝑥) = 𝑥3 − 4𝑥& + 7

3. Evaluate the given polynomial functions for the corresponding values of 𝑥.

a. 𝑓(𝑥) = 2𝑥& − 5𝑥 + 7; 𝑥 = −1 b. 𝑓(𝑥) = −3𝑥3 + 5 − 6𝑥 − 1;

𝑥 = −4 c. 𝑓(𝑥) = 3𝑥7 − 𝑥& + 7𝑥 − 5; 𝑥 =

6 d. 𝑓(𝑥) = −𝑥7 + 2𝑥3 + 𝑥& − 𝑥 +

2; 𝑥 = −3 e. 𝑓(𝑥) = 4𝑥3 + 3𝑥& − 5𝑥 + 1;

𝑥 = 3

Test Your Prerequisite Skills

• Identifying whether a given expression is a polynomial or not • Graphing polynomial functions • Evaluating polynomial functions


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STUDY GUIDE

At the end of this unit, you should be able to

• represent real-life situations using rational functions; • represent a rational function through its table of values, graph, and equation; • determine the vertical and horizontal asymptotes of a rational function; • find the domain and range of a rational function; and • solve problems involving rational functions.

Let’s go out for a trip!

Materials Needed: activity sheet, pen, and paper Instructions: 1. This activity is to be done in pairs. 2. You are to complete the given table using the information as follows.

Your family vacation is on a place 150 kilometers away from your house. You will determine the time it will take for your trip at the given rates as follows.

Objectives

Lesson 1: Introduction to Rational Functions

Warm Up!


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STUDY GUIDE

Speed (𝒙) Time (𝒕) 30 kph 40 kph 50 kph 75 kph 80 kph

Given a speed 𝑥, how will you determine your travel time? What mathematical model can you form from the data?

3. The first group who answers the activity correctly wins the game.

In the Warm Up! activity, you were able to form a mathematical model to determine the travel time for a distance of 150 km with a speed of 𝑥 kph. This situation can be expressed

as 𝑓(𝑥) = *:;-

. This function is an example of a rational function.

Note that whenever a variable appears in a polynomial, its exponent should always be a whole number.

Let us consider the function 𝑓(𝑥) = 7-. This is a rational function since both the numerator

and denominator are polynomials, and the denominator is not equal to zero. Meanwhile,

Definition 1.1: A rational function is a function

of the form 𝑓(𝑥) = <(-)=(-)

or 𝑦(𝑥) = <(-)=(-)

where 𝑃(𝑥)

and 𝑄(𝑥) are polynomials, and 𝑄(𝑥) is not equal to zero.

Learn about It!


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STUDY GUIDE

the function 𝑓(𝑥) = -B&;

is not a rational function. Observe that the numerator is a

polynomial, but the denominator is zero. Therefore, 𝑓(𝑥) = -B&;

is not a rational function.

How about the function 𝑓(𝑥) = √-B:-

? The expression in the numerator √𝑥 + 5 can be

written as (𝑥 + 5)CD. This shows that the exponent of the variable 𝑥 is not a whole number,

so we conclude that the expression is not a polynomial. Hence, 𝑓(𝑥) is not a rational function.

Example 1: Is 𝑓(𝑥) = 3

-a rational function?

Solution:

Step 1: Identify whether the numerator and the denominator are both polynomials. The numerator and the denominator are both polynomials. Step 2: Check if the denominator is not equal to zero. The denominator is not equal to zero.

Thus, 𝑓(𝑥) = 3-is a rational function.

Try It Yourself! Is 𝑓(𝑥) = -

;a rational function?

Let’s Practice!


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STUDY GUIDE

Example 2: Determine whether the function 𝑃(𝑥) = -DB:-EF-B7

is rational or not.

Solution:

Step 1: Identify whether the numerator and the denominator are both polynomials. The numerator and the denominator are both polynomials. Step 2: Check if the denominator is not equal to zero. The denominator is not equal to zero.

Hence, 𝑃(𝑥) = -DB:-EF-B7

is a rational function.

Try It Yourself!

Determine whether the function 𝑓(𝑥) = -EB3-D–:-B&

is rational or not.

Example 3: Is the function 𝑓(𝑥) = √&-B*-E

a rational function?

Solution: The expression in the numerator √2𝑥 + 1 can be written as (2𝑥 + 1)CD. This

shows that the exponent of the variable 𝑥 is not a whole number, so we conclude that the expression is not a polynomial.


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STUDY GUIDE

Try It Yourself!

Is the function 𝑓(𝑥) = -B*√7-B*

rational?

Real-World Problems Example 4: Suppose a bag company operates on a

fixed monthly cost of ₱20 000. In addition to this, the company spends ₱400 as production cost for each bag it makes. Represent the situation using a function. Is the function rational or not?

Solution: The average cost 𝐶(𝑥) can be calculated by finding the sum of the fixed

monthly cost and the variable cost, then dividing the sum by the number of bags produced.

Let 𝑥 be the number of bags the company makes. The fixed monthly cost is

20000, and the variable cost is 400𝑥. Using this information, the average cost is computed as follows.

𝐶(𝑥) =20000 + 400𝑥

𝑥

The numerator and the denominator of the function 𝐶(𝑥) = &;;;;B7;;--

are

both polynomials. Thus, the function is rational.


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STUDY GUIDE

Try It Yourself!

The length of a road is 15 km more than 50 times its width 𝑥. The length of half of the road can then be represented

as𝑓(𝑥) = :;-B*:&

. Is this function rational?

1. Complete the table below.

Is the expression on the numerator

a polynomial?

Is the expression on the

denominator a polynomial?

Is the function rational or not?

a. 𝑓(𝑥) = :-

b. 𝑓(𝑥) = -EB3-D–:-B&

c. 𝑓(𝑥) = √-B*-

d. 𝑓(𝑥) = -B:;

e. 𝑓(𝑥) = -B*7-B&

Check Your Understanding!


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STUDY GUIDE

2. Determine whether the following functions are rational or not.

a. 𝑓(𝑥) = ;-

b. 𝑓(𝑥) = -DB*-D–-B7

c. 𝑓(𝑥) = √-B7-

d. 𝑓(𝑥) = -B*;;

e. 𝑓(𝑥) = 7-B-HD

7-B&

3. The length of the road is 2 cm more than 5 times its width 𝑥. One-fourth of the

length can then be represented as𝑓(𝑥) = :-B&7

. Is the given function rational or not?


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STUDY GUIDE

Solve and Graph! Materials Needed: chalk, graphing board, paper, and pen Instructions: 1. This activity will be done per group with four members. 2. Substitute the given values of 𝑥 for finding the corresponding values for 𝑦.

𝑓(𝑥) = 2𝑥 + 1 Input (𝒙) −3 −2 −1 1 2 3 Output (𝒚) 3. Plot the points on a Cartesian Plane.

Warm Up!

Lesson 2: Representing Rational Functions through Tables, Graphs, and Equations


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STUDY GUIDE

In the Warm Up! activity, you were able to graph a linear function through solving for the values of 𝑦 with their corresponding 𝑥-values. Most of the functions you have been working on have graphs that are straight lines just like what you did in the activity. Some graphs have curved lines for quadratic or polynomial functions. These graphs represent certain equations and values when plugged into a given equation. Similarly, rational functions can also be represented using a table of values, a graph, or an equation.

Let us construct a table of values for 𝑓(𝑥) = 3-, and sketch the graph of the function.

Solution:

Step 1: Assign some values of 𝑥 as inputs. Note that 𝑥 = 0 is not included because it will make the function undefined.

Input (𝒙) −3 −2 −1 1 2 3

Output (𝒚)

Step 2: To complete the table, evaluate the function 𝑓(𝑥) = 3- for each given value of

𝑥.

Learn about It!

Definition 2.1: A table of values is composed of values (𝑥) and 𝑓(𝑥) or 𝑦 that satisfy the given function.


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STUDY GUIDE

For 𝑥 = −3:

𝑓(𝑥) =3𝑥

𝑓(−3) =3−3

𝑓(−3) = −1

For 𝑥 = −2:

𝑓(𝑥) =3𝑥

𝑓(−2) =3−2

𝑓(−2) = −1.5 Evaluating at the other values of 𝑥 results in the table as follows:

Input (𝒙) −3 −2 −1 1 2 3

Output (𝒚) −1 −1.5 −3 3 1.5 1

The value of 𝑓(𝑥) keeps decreasing for negative values of 𝑥 but suddenly becomes positive for positive values of 𝑥. Let’s observe more closely by making another table of values for the function.

Input (𝒙) 0.001 0.001 0.1 1 2 3 10 100 1000

Output (𝒚) 3000 300 30 3 1.5 1 0.3 0.03 0.003

Notice that as 𝑥 becomes closer to 0, the value of 𝑓(𝑥) increases without any bound. This will produce us the vertical asymptote of the graph.


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STUDY GUIDE

In this case, the vertical asymptote of the graph is at 𝑥 = 0. Similarly, as 𝑥 becomes larger, the value of 𝑓(𝑥) becomes smaller, eventually approaching 0. This yields the horizontal asymptote of the graph, which is 𝑦 = 0. A similar trend can be observed for the negative values of 𝑥. Can you find the asymptotes of the graph of 𝑓(𝑥) based on the following table of values?

Input (𝒙) −1000 −100 −10 −3 −2 −1 −0.1 −0.01 −0.001

Output (𝒚) −0.003 −0.03 −0.3 −1 −1.5 −3 −30 −300 −3000 Step 3: Plot the ordered pairs on a Cartesian plane, then connect them with a

smooth curve to obtain the graph of the rational function.

Definition 2.2: An asymptote is a line that a curve approaches, but does not intersect.


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STUDY GUIDE

This graph shows us the vertical and horizontal asymptotes of the function. The curves of the graph approach the lines 𝑥 = 0and 𝑦 = 0 but do not intersect them.

Example 1: Given the rational function 𝑓(𝑥) = 𝑥−3𝑥+2, construct a table of values,

and sketch the graph of the function. Solution:

Step 1: Assign some values of 𝑥 as inputs. Notice that the chosen values of 𝑥 are consecutive integers, but −2 is not included because it will result in a denominator of zero.

Input (𝒙) −7 −6 −5 −4 −3 −1 0 1 2 3

Output (𝒚)

Step 2: To complete the table, evaluate the function 𝑓(𝑥) = -F3-B&

for each given value

of 𝑥.

Input (𝒙) −7 −6 −5 −4 −3 −1 0 1 2 3 Output (𝒚) 2 2.25 2.67 3.5 6 −4 −1.5 −0.67 −0.25 0

Step 3: Plot the ordered pairs on a Cartesian plane, then connect them with a

smooth curve to obtain the graph of the rational function.

Let’s Practice!


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STUDY GUIDE

It is mentioned earlier that if 𝑥 = −2, the denominator will become 0. Thus,

the graph will never intersect the line 𝑥 = −2. This implies that the vertical asymptote of the graph is the line 𝑥 = −2.

Meanwhile, the horizontal asymptote of the function can be identified by

determining the value that 𝑓(𝑥) approaches as 𝑥 increases or decreases without bound.

The table of values below contains several negative values of 𝑥 along with the

corresponding values of 𝑓(𝑥). This function value approaches 1. Input (𝒙) −10 −100 −1000 −10000 10 100 1000 10000

Output (𝒚) 1.625 1.05 1.005 1.0005 0.58 0.95 0.995 0.9995


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Thus, the horizontal asymptote of the function is 𝑦 = 1.

Try It Yourself!

Make a table of values for 𝑓(𝑥) = 𝑥+3𝑥+1, then sketch its graph.

So far, we have determined the asymptotes of the given rational functions by substituting several values of 𝑥 and observing the trends in the resulting values of 𝑓(𝑥). However, this method is slow and unreliable because it is easy to make mistakes when evaluating complicated rational functions. For this reason, we will be using some rules that will help us identify the asymptotes of a given rational function. Remember that these rules apply to all rational functions 𝑓(𝑥) = <(-)

=(-) where 𝑃(𝑥) and 𝑄(𝑥) have no common factors. When

the numerator and the denominator have common factor(s), hole(s) at the zero(s) is(are) produced. Rules for Identifying the Asymptotes of a Rational Function Vertical asymptotes The graph of the rational 𝑓(𝑥) has vertical asymptotes at the zeros of its denominator 𝑄(𝑥). Horizontal asymptotes We can determine the horizontal asymptotes of 𝑓(𝑥) by comparing the degrees of 𝑃(𝑥) and 𝑄(𝑥). Let us call these values 𝑛 and 𝑚, respectively.

a. If 𝑛 < 𝑚, then the horizontal asymptote of 𝑓(𝑥) is the line 𝑦 = 0. b. If 𝑛 = 𝑚, then the horizontal asymptote of 𝑓(𝑥) is the line 𝑦 = PQ

RS, where the

expressions 𝑎U and 𝑏W are the leading coefficients of 𝑃(𝑥) and 𝑄(𝑥), respectively.


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STUDY GUIDE

c. If 𝑛 > 𝑚,then the graph of 𝑓(𝑥) has no horizontal asymptotes. Oblique Asymptotes If 𝑚 = 𝑛 + 1, divide the numerator by the denominator. The oblique asymptote is the quotient with the remainder ignored and set equal to 𝑦. Example 2: What is the vertical asymptote of 𝑓(𝑥) = 2

𝑥–1?

Solution:

Step 1: Reduce the rational function to lowest terms. The function is already in its lowest term. Step 2: Set the denominator to 0, and solve for 𝑥.

𝑥 − 1 = 0

𝑥 = 1

Thus, the vertical asymptote of 𝑓(𝑥) = 2𝑥–1 is 𝑥 = 1.

Try It Yourself!

What is the vertical asymptote of 𝑓(𝑥) = 𝑥–74𝑥 ?


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STUDY GUIDE

Example 3: What is the horizontal asymptote of 𝑓(𝑥) = 𝑥+1𝑥–2?

Solution: The function is already in its lowest term. Furthermore, the degree 𝑛 of the

numerator 𝑃(𝑥) and the degree 𝑚 of the denominator 𝑄(𝑥) are equal. Thus, the horizontal asymptote of 𝑓(𝑥) is the line 𝑦 = PQ

RS or 𝑦 = 1.

. Try It Yourself!

What is the horizontal asymptote of 𝑓(𝑥) = 𝑥−3𝑥+5?

Real-World Problems Example 4: You and your friends are on a summer vacation trip

that is 120 kilometers away from your school. Write a function that describes the time it takes to make the trip as a function of your speed. Create a table of values for the average speed of 40 kph, 45 kph, 50 kph, 60 kph, and 80 kph.

Solution: Recall that the product of speed and time represents distance. If 𝐷 is the

distance, 𝑟 is the speed, and 𝑡 is the time, then

𝐷 = 𝑟𝑡 If both sides of the equation are divided by 𝑟, we obtain

𝑡 =𝐷𝑟


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STUDY GUIDE

Since the location of your summer trip is 120 kilometers away from your

school, we can write the function that describes the time it takes to make the trip as a function of your speed as

𝑓(𝑟) =120𝑟

To construct the table of values for this function, evaluate the function

𝑓(𝑟) = *&;\

for each given value of the speed 𝑟.

Input (𝒙) 40 45 50 60 80

Output (𝒚) 3 2.67 2.4 2 1.5

Try It Yourself!

Sketch the graph of the function obtained in Example 4. Determine the vertical and the horizontal asymptote.


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STUDY GUIDE

1. Construct a table of values for the following rational functions, then sketch the graph.

a. 𝑓(𝑥) = 6𝑥

b. 𝑓(𝑥) = 𝑥−43

c. 𝑓(𝑥) = 𝑥+56

d. 𝑓(𝑥) = 4𝑥–7𝑥+5

2. Complete the given table. Write NONE on the corresponding cell if the asymptote

does not exist.

Function Vertical Asymptote

Horizontal Asymptote

Oblique Asymptote

a. 𝑓(𝑥) = 𝑥1

b. 𝑓(𝑥) = 𝑥−105

c. 𝑓(𝑥) = 𝑥+102

d. 𝑓(𝑥) = 𝑥2–4𝑥

e. 𝑓(𝑥) = 2𝑥−1𝑥−2

3. The first location in your school’s field trip is 100 kilometers away from your school. Write a function that describes the time it takes to make the trip as a function of the speed of the bus. Create a table of values for the average speed of 50 kph, 55 kph, 60 kph, and 70 kph. Sketch the graph of the function, and determine the vertical and the horizontal asymptote.



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STUDY GUIDE

Locate the ends! Materials Needed: graphing papers, blue and orange color pens

Instructions:

1. This activity will be done by pairs. 2. Your teacher will hand you four different kinds of graph. 3. Locate the farthest left and right points. Mark them, and color the 𝑥 −axis

blue. 4. Identify the lowest and highest points and mark the corresponding 𝑦 −axis

orange. The graphs are shown as follows:

Warm Up!

Lesson 3: Domain and Range of a Rational Function


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STUDY GUIDE

The concept of domain and range is important as you learn how to graph rational functions. The activity in Warm Up! gave you a glimpse about the domain and range

Learn about It!


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STUDY GUIDE

given a graph. The blue marked 𝑥 −axis indicates the domain while the orange marked 𝑦 −axis is the range. These graphs, however, do not represent rational functions. So how then do we determine the domain and range of a rational function? In Lesson 2, you have learned how to represent functions through table of values,

graphs, and equations. Recall that the function 𝑓(𝑥) = 3- can be represented by the

graph as follows. If we inspect the graph, we find that the function has a vertical asymptote at 𝑥 = 0, which implies that the graph does not contain a point for which 𝑥 = 0. Therefore, the domain of this function is the set of all real numbers 𝑥 such that 𝑥 ≠ 0. In symbols, we write this as 𝐷: {𝑥|𝑥 ≠ 0}.

Definition 3.1: The domain of a function is the set of all values of 𝑥 that have corresponding values of 𝑦. It contains all values that go into the function.


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STUDY GUIDE

Similarly, the values of 𝑦 are all nonzero real numbers. Therefore, the range of the function is the set of all real numbers except 𝑦 = 0. In symbols, we write this as 𝑅: {𝑦|𝑦 ≠ 0}. Remember that the domain and range of a function are obtained by finding and excluding the appropriate restrictions. In the case of rational functions, these restrictions can be found at the vertical and horizontal asymptotes because there are no points plotted along them. To find the domain of a rational function, perform the following steps:

1. Make sure that the fraction is in its simplest form. When the numerator and the denominator have common factor(s), hole(s) at the zero(s) is(are) produced.

2. Find the vertical asymptotes of the function by setting the denominator equal to zero and solving the resulting equation.

3. Exclude the asymptote/s obtained in the previous step from the set of real numbers. Reject the zero as well of the common factor that is canceled. The remaining elements of the set of real numbers comprise the domain of the function.

A similar procedure can be used to determine the range of a rational function:

1. Make sure that the fraction is in its simplest form. 2. Find the horizontal asymptotes of the given rational function by comparing the

degrees of the numerator and the denominator.

Definition 3.2: The range of a function is the set of all values of 𝑦 that can be obtained from the possible values of 𝑥. It contains all possible values of the function.


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STUDY GUIDE

3. Exclude the asymptote(s) obtained in the previous step from the set of real numbers. If there is a common factor canceled, say (𝑥 − 𝑘), reject also the value for 𝑓(𝑘). The remaining elements of the set of real numbers comprise the range of the function.

Example 1: Determine the domain and range of 𝑓(𝑥) = 𝑥−3𝑥+2.

Solution:

Step 1: Reduce the rational function to its simplest form. The function is already in its simplest form.

For the domain of the function:

Step 2.a: Find the vertical asymptotes of the function by setting the denominator equal

to zero, and solving the resulting equation.

𝑥 + 2 = 0𝑥 = −2

Step 2.b: Exclude the asymptote/s obtained in the previous step from the set of real

numbers. The remaining elements of the set of real numbers comprise the domain of the function.

Thus, the domain of this function is the set of all real numbers 𝑥 such that 𝑥 ≠ −2. In symbols, 𝐷: {𝑥|𝑥 ≠ −2}.

Let’s Practice!


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STUDY GUIDE

For the range of the function: Step 3.a: Find the horizontal asymptotes of the given rational function by comparing

the degrees of the numerator and the denominator.

The numerator and the denominator of the function have the same degree. Recall that if 𝑛 = 𝑚, then the horizontal asymptote of 𝑓(𝑥) is the line 𝑦 = PQ

RS,

where the expressions 𝑎U and 𝑏W are the leading coefficients of 𝑃(𝑥) and 𝑄(𝑥), respectively. For the given rational function, this expression is equal to 1.


numbers. The remaining elements of the set of real numbers comprise the range of the function.

Thus, the range of this function is the set of all real numbers 𝑦 such that 𝑦 ≠ 1. In symbols, 𝑅: {𝑦|𝑦 ≠ 1}.

Try It Yourself!

Determine the domain and range of 𝑓(𝑥) = 2𝑥−1𝑥−4 .

Example 2: Find the domain and range of 𝑓(𝑥) = (𝑥−1)(𝑥+2)

(𝑥−1)(𝑥+3).

Solution:

Step 1: Reduce the rational function to its simplest form.


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STUDY GUIDE

The numerator and denominator have a common factor of 𝑥 − 1. Recall that when the numerator and the denominator have common factor(s), hole(s) at the zero(s) is(are) produced. Hence, the simplified form of the function is

𝑓(𝑥) = (𝑥+2)(𝑥+3).


Step 2.a: Find the vertical asymptotes of the function by setting the denominator equal

to zero and solving the resulting equation.

𝑥 + 3 = 0𝑥 = −3


numbers. Reject the zero as well of the common factor that is canceled. Equating the common factor 𝑥 − 1 to zero yields to 𝑥 = 1. This is not included

in the domain of the function. The remaining elements of the set of real numbers comprise the domain of

the function.

Thus, the domain of this function is the set of all real numbers 𝑥 such that 𝑥 ≠ −3, 1. In symbols, 𝐷: {𝑥|𝑥 ≠ −3, 1}.




30

STUDY GUIDE


RS,



numbers. If there is a common factor canceled, say (𝑥 − 𝑘), reject also the value for 𝑓(𝑘). The remaining elements of the set of real numbers comprise the range of the function.

The common factor canceled is 𝑥 − 1. Hence, we reject also the value for 𝑓(1).

𝑓(𝑥) =(𝑥 + 2)(𝑥 + 3)

𝑓(1) =(1 + 2)(1 + 3)

𝑓(1) =34

Thus, the range of this function is the set of all real numbers 𝑦 such that

𝑦 ≠ 1, 37. In symbols, 𝑅: e𝑦|𝑦 ≠ 1, 3

7f.

Try It Yourself!

Determine the domain and range of 𝑓(𝑥) = (𝑥−3)(𝑥+4)(𝑥+4)(𝑥−1).


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STUDY GUIDE

Example 3: Determine the asymptote of the function 𝑓(𝑥) = 𝑥2+4𝑥−12𝑥2−4 .

Solution:

Step 1: Reduce the rational function to its simplest form.

𝑥& + 4𝑥 − 12𝑥& − 4 =

(𝑥 − 2)(𝑥 + 6)(𝑥 + 2)(𝑥 − 2)

𝑥& + 4𝑥 − 12𝑥& − 4 =

(𝑥 + 6)(𝑥 + 2)

The simplified form of the function is 𝑓(𝑥) = (𝑥+6)

(𝑥+2).


Step 2.a: Find the vertical asymptotes of the function by setting the denominator equal to zero and solving the resulting equation.

𝑥 + 2 = 0

𝑥 = −2

Step 2.b: Exclude the asymptote/s obtained in the previous step from the set of real numbers. Reject the zero as well of the common factor that is canceled.

Equating the common factor 𝑥 − 2 to zero yields to 𝑥 = 2. This is not included

in the domain of the function. The remaining elements of the set of real numbers comprise the domain of

the function.


32

STUDY GUIDE

Thus, the domain of this function is the set of all real numbers 𝑥 such that 𝑥 ≠ −2, 2. In symbols, 𝐷: {𝑥|𝑥 ≠ −2, 2}.




RS,



numbers. If there is a common factor canceled, say (𝑥 − 𝑘), reject also the value for 𝑓(𝑘). The remaining elements of the set of real numbers comprise the range of the function.

The common factor canceled is 𝑥 − 2. Hence, we reject also the value for 𝑓(2).

𝑓(𝑥) =(𝑥 + 6)(𝑥 + 2)

𝑓(2) =(2+ 6)(2 + 2)

𝑓(2) =84 = 2

Thus, the range of this function is the set of all real numbers 𝑦 such that 𝑦 ≠ 1, 2. In symbols, 𝑅: {𝑦|𝑦 ≠ 1, 2}.


33

STUDY GUIDE

Try It Yourself!

Determine the domain and range of 𝑓(𝑥) = 𝑥2+3𝑥−18𝑥2+𝑥−12 .

Real-World Problems Example 4: Suppose a driver goes to a gas station to refuel.

He noticed that the station sells gasoline at ₱40 per liter, and he has ₱500 pesos in his wallet. Represent the number of liters of gasoline that the driver can purchase. What is the domain of the given function?

Solution: The number of liters of gasoline that the driver can purchase can be

represented by the function 𝑓(𝑥) = -7;

, where 𝑥 is the amount of money that

the driver will spend. This gives us certain restrictions based on the amount that the driver has on hand. Therefore, we can only input values from 0 to 500 into the function. The domain of the function is 𝐷:{𝑥|0 ≤ 𝑥 ≤ 500}.

Try It Yourself!

Records from a factory producing electronic components show that on average, new employees can assemble 𝑁(𝑡) components per day after 𝑡 days of training, where

𝑁(𝑡) = i:jjB:

; 𝑡 ≥ 0. What is the domain and range of the

function?


34

STUDY GUIDE

1. Compute for the domain and range of the following rational functions.

Function Domain Range a. 𝑓(𝑥) = 3

𝑥

b. 𝑓(𝑥) = 𝑥−62

c. 𝑓(𝑥) = 𝑥+52

d. 𝑓(𝑥) = 𝑥2+93

e. 𝑓(𝑥) = 𝑥2+𝑥−906

2. Solve for the vertical and horizontal asymptote of the following functions. Show

your complete solutions.

Function Vertical Asymptote

Horizontal Asymptote

a. 𝑓(𝑥) = 6𝑥

b. 𝑓(𝑥) = 𝑥−43

c. 𝑓(𝑥) = 𝑥+37

d. 𝑓(𝑥) = 𝑥2+62

e. 𝑓(𝑥) = 𝑥2+𝑥−1005

3. After a drug is injected into a patient’s bloodstream, the concentration 𝐶 of the drug

in the bloodstream in 𝑡 minutes after the injection is given by 𝐶(𝑡) = &;jjDB&

; 𝑡 ≥ 0.

What is the domain and range of the given function?



35

STUDY GUIDE

Boards Up! Materials Needed: drill boards, whiteboard marker, pen, and paper Instructions: 1. This game is to be in groups with 3 members each. 2. Your teacher will present 5 different rational functions. 3. For every rational function given, the first member of the group will evaluate 3

different values of 𝑥. These values are the first 3 multiples of 5. 4. The second member will determine the asymptotes of the rational functions. 5. The third member will identify the domain and range of the function. 6. The first group who completed the answers correctly gets the point.

The game in the Warm Up! activity allowed you to recall different concepts you have learned in this unit. Let us walk through these concepts again.

Recall that a rational function is of the form 𝑓(𝑥) = <(-)=(-)

, where 𝑃(𝑥) and 𝑄(𝑥) are both

polynomials and 𝑄(𝑥) ≠ 0. A rational function may also be represented using a table of values, a graph, or an equation.

Lesson 4: Solving Problems Involving Rational Functions

Learn about It!

Warm Up!


36

STUDY GUIDE

You have also learned about asymptotes. Recall that an asymptote is a line that a curve approaches, but does not intersect. There are rules that we follow to determine the asymptotes of the graph of a rational function. Vertical asymptotes The graph of the rational 𝑓(𝑥) has vertical asymptotes at the zeros of its denominator 𝑄(𝑥). Horizontal asymptotes We can determine the horizontal asymptotes of 𝑓(𝑥) by comparing the degrees of 𝑃(𝑥) and 𝑄(𝑥). Let’s call these values 𝑛 and 𝑚, respectively.

a. If 𝑛 < 𝑚, then the horizontal asymptote of 𝑓(𝑥) is the line 𝑦 = 0. b. If 𝑛 = 𝑚, then the horizontal asymptote of 𝑓(𝑥) is the line 𝑦 = PQ

RS, where the

expressions 𝑎U and 𝑏W are the leading coefficients of 𝑃(𝑥) and 𝑄(𝑥), respectively. c. If 𝑛 > 𝑚,then the graph of 𝑓(𝑥) has no horizontal asymptotes.

Oblique Asymptotes If 𝑚 = 𝑛 + 1, divide the numerator by the denominator. The oblique asymptote is the quotient with the remainder ignored and set equal to 𝑦. Moreover, you have learned how to identify the domain and range of a function. The domain of a function is the set of all values of 𝑥 that have corresponding values of 𝑦. It contains all the valid inputs of the function. On the other hand, the range of a function is the set of all values of 𝑦 that can be obtained from the possible values of 𝑥. It contains all the valid outputs of the function. Using these concepts, let us try to solve some word problems involving rational functions.


37

STUDY GUIDE

Example 1: Suppose a company that manufactures pencils in Quezon City operates on a

fixed monthly cost of ₱10 000. In addition to this, the company spends ₱5 as production cost for each pencil. Represent the situation using rational functions.

Solution: The average cost 𝐶(𝑥) can be calculated by finding the sum of the fixed

monthly cost and the variable cost, then dividing the sum by the number 𝑥 of pencils produced. If the fixed monthly cost is ₱10 000 and the variable cost is ₱5, then the situation can be represented as

𝐶(𝑥) = 10000+5𝑥𝑥 .

Try It Yourself! Suppose a company that manufactures bags operate on a fixed monthly cost of ₱15 000. In addition to this, the company spends ₱150 as production cost for each bag it makes. Represent the situation using rational functions.

Example 2: To join a tennis club, you have to pay a membership fee of ₱2 000, plus ₱100

per session. What is the average cost function and the average cost per session if you attend 10 sessions?

Solution: The average cost function can be expressed as the sum of the membership

fee and the variable cost, divided by the number of sessions. Let 𝑥 be the number of sessions. If the fixed monthly cost is ₱2 000 and the variable cost is ₱100, then the average cost function is

Let’s Practice!


38

STUDY GUIDE

𝐶(𝑥) = 2000+100𝑥𝑥 .

To determine the average cost if you attend 10 sessions, substitute 𝑥 = 10 into the average cost function.

𝐶(𝑥) =2000 + 100𝑥

𝑥

𝐶(10) =2000 + 100(10)

10

𝐶(10) =2000 + 1000

10

𝐶(10) =300010

𝐶(10) = 300

Therefore, the average cost over 10 sessions is ₱300.

Try It Yourself!

To join a golf club, you have to pay a membership fee of ₱4 000, plus ₱200 per session. What is the average cost function and the average cost per session if you attend 5 sessions?

Example 3: Suppose you are staying at a hotel that charges ₱7 000 upon check-in, plus

₱1000 daily. What is the average cost function and the average daily cost if you stay there for 7 days?

Solution: The average cost function can be expressed as the sum of the check-in fee

and the variable cost, divided by the number of days. Let 𝑥 be the number of days. If the fixed monthly cost is ₱7 000 and the variable cost is ₱1000, then the average cost function is


39

STUDY GUIDE

𝐶(𝑥) = 7000+1000𝑥𝑥 .

To determine the average cost if you attend 7 days, substitute 𝑥 = 7 into the average cost function.

𝐶(𝑥) =7000 + 1000𝑥

𝑥

𝐶(7) =7000 + 1000(7)

7

𝐶(7) =7000 + 7000

7

𝐶(7) =140007

𝐶(7) = 2000

Therefore, the average daily cost of a seven-day hotel stay is ₱2000.

Try It Yourself! Suppose you are staying at a hotel that charges ₱5 500 upon check-in, plus ₱500 daily. What is the average cost function and the average daily cost if you stay there for 10 days?

More Real-World Problems Example 4: A car rental company charges a

fixed price of ₱5000. For every kilometer traveled, an additional ₱20 will be charged. What is the average cost per kilometer if you rent a car and drive it for 25 kilometers? Assume that the fuel, maintenance, and other relevant costs will not be charged to you.


40

STUDY GUIDE

Solution: The average cost 𝐶(𝑥) can be calculated by adding the fixed rental cost and

the additional charge per kilometer, then dividing the sum by the number of kilometers traveled.

𝐶(𝑥) =5000 + 20𝑥

𝑥

To determine the average cost per stay, substitute 𝑥 = 25 to the average cost function.

𝐶(𝑥) =5000 + 20𝑥

𝑥

𝐶(25) =5000 + 20(25)

25

𝐶(25) =5000 + 500

25

𝐶(25) =550025

𝐶(25) = 220

Therefore, the average rental cost per kilometer for a 25-kilometer trip is ₱220.

Try It Yourself! A van rental company charges a fixed ₱4 000 fee, plus ₱30 per kilometer traveled. What is the average cost per kilometer if you rent a car and drive it 10 kilometers? Assume that the fuel, maintenance, and other relevant costs will not be charged to you.


41

STUDY GUIDE

Read and analyze the following problems. Show your solutions. 1. Suppose a company that manufactures teddy bear operates on a fixed monthly

cost of ₱200 000. In addition to this, the company spends ₱150 as production cost for each teddy bear it makes. Represent the situation using rational functions. What will be the average cost per teddy bear that will be produced?

2. Suppose you are staying at a hotel that charges ₱3 000 upon check-in, plus ₱200 daily. What is the average cost function and the average daily cost if you stay there for 12 days?

3. A van rental company charges a fixed ₱3 000 fee, plus ₱20 per kilometer traveled. What is the average cost per kilometer if you rent a car and drive it for 15 kilometers? Assume that the fuel, maintenance, and other relevant costs will not be charged to you.

4. Suppose you rent a bike and you need to pay an initial fee of ₱100 and another ₱15 per hour. How much will you spend for a total of 4 hours? What is the average cost per hour?

1. Suppose you are given a rational function 𝑓(𝑥). Is the reciprocal of 𝑓(𝑥) always a rational function? Support your answer with an example.

Challenge Yourself!



42

STUDY GUIDE

2. Is it possible for a function to have more than one vertical asymptote? Describe how this can be done, or explain why it is not possible.

3. Given the rational function 𝑓(𝑥) = -B:-DFl

, what should be the value of 𝑐 so that the graph of the function has a hole at 𝑥 = −5?

4. Suppose you are asked to write a rational function 𝑓(𝑥) that has a vertical

asymptote at 𝑥 = 3, a horizontal asymptote at 𝑦 = 1, and zero at 𝑥 = 5, what will be the function?

5. What is the difference between the functions f(x) = *;p

and f(x) = √pB:pif 𝑥 = 2?

You are a research analyst of a company that is about to venture a new business. You are to choose 3 different business ventures and explore the average cost function of the product they produce. Create a presentation of these functions which includes the following:

a. at least 5 sample number of products and represent these data through a table of values, graphs, and equations

b. domain and range of the functions c. asymptotes of the graph

Present the data to the research supervisor. He/she will assess the data presentation using the rubrics as follows.

Performance Task


43

STUDY GUIDE

Performance Task Rubric

Criteria Below

Expectation (0–49%)

Needs Improvement

(50–74%)

Successful Performance

(75–99%)

Exemplary Performance

(99+%)

Punctuality and Neatness

The data is submitted late and has many erasures.

The data is submitted late and has few erasures.

The data is submitted on time but has few erasures.

The data is submitted on time and has no erasure.

Accuracy of Computations

There are significant errors in computations that lead to the wrong relativity of data.

There are few errors in the computation, but there is no clear reference in the analysis of the data.

All computations are correct, and the data are analyzed properly.

All computations are correct and with a complete solution. The data are analyzed with clear references.

Presentation

The students did not present the output.

The students presented the output but are not confident with their work.

The student presented the data and are confident with their work.

The students presented the data and are very confident with their work.


44

STUDY GUIDE

Key Terms & Definitions

Key Terms Definitions

Rational Function

A rational function is a function of the form 𝑓(𝑥) = <(-)=(-)

or

𝑦(𝑥) = <(-)=(-)

where 𝑃(𝑥) and 𝑄(𝑥) are polynomials, and 𝑄(𝑥) is

not equal to zero.

Table of Values A table of values is composed of values (𝑥) and 𝑓(𝑥) or 𝑦 that satisfy the given function.

Asymptote An asymptote is a line that a curve approaches, but does not intersect.

Domain The domain of a function is all the set of all values of 𝑥 that have corresponding values of 𝑦. It contains all values that go into the function.

Range The range of a function is the set of all values of 𝑦 that can be obtained from the possible values of 𝑥. It contains all possible values of the function.

Wrap-up

can be represented through

Rational Functions

graph table of values equation


45

STUDY GUIDE

Lesson 1

1. No. The denominator is zero. 2. Yes, it is a rational function. 3. No, the function is not rational. 4. Yes, it is a rational function.

Lesson 2

1. 𝑓(𝑥) = 𝑥+3𝑥+1

𝒙 −3 −2 0 1 2 3 4

𝒇(𝒙) 0 −1 3 2 1.67 1.5 1.4

2. 𝑥 = 0 3. 𝑦 = 1

Key to Let’s Practice!


46

STUDY GUIDE

4. Graph:

Vertical Asymptote: 𝑥 = 0 Horizontal Asymptote: 𝑦 = 0

Lesson 3:

1. 𝐷: {𝑥|𝑥 ≠ 4} 𝑅: {𝑦|𝑦 ≠ 2}

2. 𝐷: {𝑥|𝑥 ≠ −4, 1} 𝑅: {𝑦|𝑦 ≠ 1, 1.4}

3. 𝐷: {𝑥|𝑥 ≠ −4, 3} 𝑅: {𝑦|𝑦 ≠ 1, 1.29}

4. 𝐷:{𝑥|𝑥 ≥ 0} 𝑅: {𝑦|0 ≤ 𝑦 < 75}

Lesson 4: 1. 𝐶(𝑥) = *:;;;B*:;-

-

2. 𝐶(𝑥) = 7;;;B&;;--

; ₱1 000


47

STUDY GUIDE

3. 𝐶(𝑥) = ::;;B:;;--

; ₱1 050 4. 𝐶(𝑥) = 7;;;B3;-

-; ₱430

Purplemath. “Adding and Subtracting Fractions.” Accessed April 21, 2018.

http://www.purplemath.com/modules/fraction4.htm Monterey Institute. “Finding Domain and Range.” Accessed April 21, 2018. http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_

RESOURCE/U17_L2_T3_text_final.html

References

GM 04 Q1 FD rev - d14fikpiqfsi71.cloudfront.net

Documents

Transcript of GM 04 Q1 FD rev - d14fikpiqfsi71.cloudfront.net