GENERAL MATHEMATICSpauljorelsantos.weebly.com/uploads/9/2/9/2/92921742/... · 2020. 8. 15. · Learning Module for General Mathematics 2 PARTS OF THE MODULE • Expectations - These

Learning Module for General Mathematics

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Republic of the Philippines Department of Education National Capital Region

DIVISION OF CITY SCHOOLS – MANILA Manila Education Center Arroceros Forest Park

Antonio J. Villegas St. Ermita, Manila

GENERAL MATHEMATICS

Quarter 1 Week 4 Module 9 Learning Competency:

Represents real-life situations using one-to-one functions. (M11GM-Id-1)


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Before starting the module, I want you to set aside other tasks that will disturb you while enjoying the lessons. Read the simple instructions below to successfully enjoy the objectives of this kit. Have fun! 1. Follow carefully all the contents and

instructions indicated in every page of this module.

2. Write on your notebook the concepts about the lessons. Writing enhances learning, that is important to develop and keep in mind.

3. Perform all the provided activities in the module.

4. Let your facilitator/guardian assess your answers using the answer key card.

5. Analyze conceptually the posttest and apply what you have learned.

6. Enjoy studying!

HOW TO USE THIS MODULE?


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PARTS OF THE MODULE

• Expectations - These are what you will be able to know after completing the lessons in the module.

• Pre-test - This will measure your prior knowledge and the concepts to be mastered throughout the lesson.

• Looking Back to your Lesson - This

section will measure what learnings and skills did you understand from the previous lesson.

• Brief Introduction- This section will give

you an overview of the lesson. • Activities - This is a set of activities you

will perform with a partner. • Remember - This section summarizes the

concepts and applications of the lessons. • Check your Understanding - It will verify

how you learned from the lesson.

• Post-test - This will measure how much you have learned from the entire module


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One-to-One Functions

You will represent real-life situations using one-to-one functions.

Specifically, this module will help you to:

Define one-to-one functions; Determine whether the given relation is a function; Determine the given graph whether a one-to-one function or not; and Apply horizontal line test.

Read the questions carefully. Encircle the letter of the correct answer.

1. It is a function in which for each value of y in the range of f, there is just one value x in the domain of f such that y = f(x). A. Constant Function C. Linear Function B. Identity Function D. One-to-one Function

2. Which of the following is a one-to-one function? A. Books to authors C. True or False questions to answers B. SIM cards to cell phone numbers D. Real number to its square.

3. Which of the following is NOT a one-to-one function? A. {(0, 0), (1, 1), (2, 8), (3, 27), (4, 64)} B. {(-2, 6), (-1, 3), (0, 2), (1, 5), (2, 8)} C. {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)} D. {(0, 4), (1, 5), (2, 6), (3, 7), … (n, n+4), …)}

4. A one-to-one function crosses a horizontal line how many times? A. 0 C. 2 B. 1 D. 3

5. All of the following are not one-to-one functions, EXCEPT: A. 𝑦𝑦 = 𝑥𝑥3 + 2 C. 𝑦𝑦 = |𝑥𝑥 + 1| B. 𝑦𝑦 = 𝑥𝑥2 + 2𝑥𝑥 − 1 D. 𝑦𝑦 − 𝑥𝑥4

Before going further, let us try to recall about relation. A relation is a rule that relates values from a set of values (called the domain) to a second set of values (called the range). A function is a set of ordered pairs (x, y) such that no two ordered pairs have the same x-value but different y – values. The domain of a relation is the set of first coordinates and the range is the set of second coordinates.

LESSON 9

EXPECTATIONS

Let us start your journey in learning more on one-to-one functions. I am sure you are ready and excited to answer the Pretest. Smile and Enjoy! PRETEST

Great, you finished answering the questions. You may request your facilitator to check your work. Congratulations and keep on learning! LOOKING BACK TO YOUR LESSON


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1. Find the domain and range of the given ordered pairs. A. f = {(1, 2), (2, 2), (3, 5), (4, 5)} B. 𝑔𝑔 = {(1, –1), (2, –3), (0, 5), (–1, 3), (4, –5), (–1, 5), (4, –4)} C. ℎ = {(10, 9), (– 2, – 16), (– 6, 7), (5, 8), (8, – 16), (– 11, 9)}

2. Which of the following statements represents a function? A. Students to their current age. B. Countries to its capital. C. Destination versus tricycle fare.

3. Determine which of the equations define a function. A. 𝑦𝑦 = 3𝑥𝑥 + 2 B. 𝑦𝑦 = √𝑥𝑥 + 2 C. 𝑥𝑥2 + 𝑦𝑦2 = 4

4. Indicate whether each graph is the graph of a function.

A. B. C.

Functions are described as mapping of its domain to its range. Recall that a relation is one-to-one if and only if each element of its domain corresponds to a unique element in its range, and each element of its range

corresponds to a unique element of its domain. The following real-life situations illustrate important type of function the one-to-one function.

1. DepEd is developing a system of identification for all learners of the Philippines. This is the Learner’s Identification Number (LIS) System that aims to provide a unique LIS to every learner. Its aim to ensure that no two LIS is assigned to a Filipino learner, and that no two Filipinos learners have the same LIS.

2. One of the primary moral values that is advocated and taught by the Catholic Church is the blessedness of marriage vow. It aims to promote happy marriage between a Living Catholic man and a Catholic woman who have entered into a marriage contract, that is, one-man-one-woman relationship. These are just some of the situations that use the concept of one-to-one relationships.

One-to-One Function

Definition: The function 𝑓𝑓 is one-to-one if for any 𝑥𝑥1, 𝑥𝑥2 in the domain of 𝑓𝑓, then 𝑓𝑓(𝑥𝑥1) ≠ 𝑓𝑓(𝑥𝑥2). That is, the same y – value is never paired with two different 𝑥𝑥 −𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣.

BRIEF INTRODUCTION


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The arrow diagrams below shown an example and non-example of one-to-one function.

Domain Range Domain Range

One-to-one Function Not One-to-one Function

Relation f is one-to-one because there are no two distinct inputs that correspond to the same output. Relation 𝑔𝑔 is not one-to-one because two different inputs, 0 and 8, have the same output of 4.

The function 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 − 7 is one-to-one because if 𝑥𝑥1 and 𝑥𝑥2 are real numbers

such that 𝑓𝑓(𝑥𝑥1) = 𝑓𝑓(𝑥𝑥2), then

2𝑥𝑥1 − 7 = 2𝑥𝑥2 − 7

2𝑥𝑥1 = 2𝑥𝑥2

𝑥𝑥1 = 𝑥𝑥2

The functions 𝑥𝑥, 𝑥𝑥3, 𝑥𝑥5, 1𝑥𝑥

, 𝑣𝑣𝑒𝑒𝑒𝑒. Are one-to-one because 𝑥𝑥1 ≠ 𝑥𝑥2, then 𝑥𝑥31 ≠ 𝑥𝑥32,

𝑥𝑥51 ≠ 𝑥𝑥52, 1𝑥𝑥1≠ 1

𝑥𝑥2.

Function 𝑥𝑥2, 𝑣𝑣𝑎𝑎𝑎𝑎 𝑥𝑥6 are not one-to-one because (-1)2 = (1)2 and (-1)6 = (1)6.

Graphically, a function can be easily identified as one-to-one using the horizontal line test.

Horizontal Line Test

A function is one-to-one if each horizontal line does not intersect the graph at more than one point.

(a) (b) (c)

one-to-one not one-to-one one-to-one

2

4

6

8

1

3

5

7

f -5

0

8

15

1

4

9

g

Therefore, 𝑓𝑓 is one-to-one.


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Example 1. Determine whether each function is a one-to-one function.

(a) 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 − 5 (b) 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 3

Solution.

(a) 𝑓𝑓(𝑥𝑥1) = 𝑓𝑓(𝑥𝑥2) (b) 𝑓𝑓(𝑥𝑥1) = 𝑓𝑓(𝑥𝑥2)

2𝑥𝑥1 − 5 = 2𝑥𝑥2 − 5 𝑥𝑥21 + 3 = 𝑥𝑥22 + 3

𝑥𝑥1 = 𝑥𝑥2 𝑥𝑥21 = 𝑥𝑥22

∴ 𝑓𝑓 is one-to-one function �𝑥𝑥21 = �𝑥𝑥21

±𝑥𝑥1 = ±𝑥𝑥2

∴ 𝑓𝑓 is not one-to-one because 𝑓𝑓(2) = 𝑓𝑓(−2) = 7

Example 2. Determine whether each graph represents a one-to-one function.

(a) (b)

Solution.

Use horizontal line test to answer each.

(a) (b)

Example 3. Determine whether the given relation is a function, if it is a function, determine whether it is one-to-one.

(a) The relation pairing SSS member to his or her SSS number.

(b) The relation pairing a real number to its square.

(c) The relation pairing to his or her citizenship.

Solution.

(a) Each SSS member assigned a unique SSS number, thus the relation is a function. Further, two different members cannot be assigned the same SSS number. Thus, the function is one-to-one.

not one-to-one function, the

horizontal line intersects the graph

at two points.

one-to-one function, any horizontal line

intersects the graph at exactly one point.


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(b) Each real number has a unique perfect square. Thus, the relation is a function. However, two different real numbers such as 2 and -2 may have the same square. Thus, the function is not one-to-one.

(c) The relation is a function because a person can have dual citizenship.

Activity 1. Determine whether each function is one-to-one or not.

1. (𝑥𝑥) = 𝑥𝑥 + 5

2. 𝑔𝑔(𝑥𝑥) = 𝑥𝑥4 + 3

3. ℎ(𝑥𝑥) = 1𝑥𝑥−2

4. 𝑝𝑝(𝑥𝑥) = |2𝑥𝑥 − 7|

5. 𝑒𝑒(𝑥𝑥) = 𝑥𝑥8 − 2

6. 𝑓𝑓(𝑥𝑥) = 3 − 𝑥𝑥2

7. 𝑚𝑚(𝑥𝑥) = √𝑥𝑥2 − 4

8. 𝑣𝑣(𝑥𝑥) = (𝑥𝑥 + 2)2

9. 𝑣𝑣(𝑥𝑥) = 𝑥𝑥

10. 𝑣𝑣(𝑥𝑥) = 𝑥𝑥2 − 𝑥𝑥4

Activity 2. The graph 𝑓𝑓 is given. Determine whether 𝑓𝑓 is one-to-one.

1. 6.

2. 7.

3. 8.

ACTIVITIES


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4. 9.

5. 10.

Activity 3. Which of the following are one-to-one functions?

1. A school ID for a student.

2. A key to its padlock. 3. Books to authors

4. SIM cards to cell phone numbers

5. A keyboard for a computer.

6. True or False questions to answers.

7. The height of a person at a given time in their life.

8. The relations pairing an airport to its airport code.

9. Response of your friends when answering “How are you?”.

10. The relation pairing a distance d (in kilometers) traveled along a

given jeepney route to the jeepney fare for traveling that distance.

A one-to-one function is a function in which for each value of 𝑦𝑦 in the range of 𝑓𝑓, there is just one value of x in the domain of 𝑓𝑓 such that 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). For example, 𝑓𝑓 is one-to-one if 𝑓𝑓(𝑥𝑥1) = 𝑓𝑓(𝑥𝑥2) implies 𝑥𝑥1 = 𝑥𝑥2. In general,

• 𝑓𝑓(𝑥𝑥) = 𝑣𝑣𝑥𝑥 − 𝑏𝑏, 𝑣𝑣 ≠ 0, is one-to-one. • 𝑓𝑓(𝑥𝑥) = 𝑥𝑥𝑛𝑛, 𝑖𝑖𝑓𝑓 𝑎𝑎 𝑖𝑖𝑣𝑣 𝑣𝑣𝑣𝑣𝑣𝑣𝑎𝑎, it is not one-to-one. • 𝑓𝑓(𝑥𝑥) = 𝑥𝑥𝑛𝑛, 𝑖𝑖𝑓𝑓 𝑎𝑎 𝑖𝑖𝑣𝑣 𝑜𝑜𝑎𝑎𝑎𝑎, it is one-to-one. • 𝑓𝑓(𝑥𝑥) = 𝑥𝑥−𝑛𝑛, 𝑖𝑖𝑓𝑓 𝑎𝑎 𝑖𝑖𝑣𝑣 𝑜𝑜𝑎𝑎𝑎𝑎, it is one-to-one

A function is one-to-one if no horizontal line does not intersect the graph at more than one point.

REMEMBER


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The table relates by place of

residence in the Philippines, 𝑥𝑥 to the

number of COVID-19 patients 𝑦𝑦 as

of April 15, 2020. Does this relation

define a one-to-one function?

Read the questions carefully. Encircle the letter of the correct answer.

1. Which of the following function is not one-to-one? A. {(0, 1), (1, 2), (2, 3), (3, 4)} C. {(0, 1), (1, 0), (2, 3), (3, 2)} B. {(0, 0), (1, 1), (2, 2), (3, 3)} D. {(0, 1), (1, 0), (2, 0), (3, 2)}

2. Which function is one-to-one? A. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥4 + 3 C. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| + 3 B. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 3 D. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 2𝑥𝑥 + 1

3. All of the following graph is one-to-one function, EXECPT. A. C. B. D.

4. A method of determining whether or not a graph represent a one-to-one function.

A. Horizontal Line Test C. Square Root Method B. Vertical Line Test D. Piecewise Function

Place of Residence (𝑥𝑥)

Number of cases (𝑦𝑦)

City of Manila 470

City of Pasig 229

Quezon City 992

Cebu City 32

Davao City 80

Baguio City 17

Caloocan City 121

Imus City 32

Batangas City 19

Bulacan 8

CHECK YOUR UNDERSTANDING

POSTTEST

https://www.statista.com/statistics/1103623/philippines-coronavirus-covid-19-cases-by-residence/


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5. Temperature readings T (in 0C) were recorded every three hours from midnight until 6 PM. The time 𝑒𝑒 was measured in hours from midnight.

Time 0 3 6 9 12 15 18 T(in 0C) 24 26 28 30 32 30 28

A. 22 C. 26 B. 24 D. 28

REFLECTIVE LEARNING SHEET

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

Due to high technology, numerous ways of falsifying public documents and even fake accounts in social media has been a public issue these days. It’s a good that the Department of Education has its own way of identifying students’ information through Learner’s Reference Number (LRN). Each student has its own LRN given to them since Kindergarten, where no students should have the same LRN.

This situation shows a one-to-one function. Provide at least five real – world situation or scenario that can be modeled by a one-to-one function.


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To further explore the concept learned today and if it possible to connect the internet, you may visit the following links:

https://www.youtube.com/watch?v=nE_ykVScQww&t=3s

Oronce, O. A. (2016). General Mathematics (First Edition). Rex Book Store Inc.

Orines, F. B. (2016). Next Century Mathematics 11 General Mathematics. Phoenix Publishing House.

https://study.com/academy/practice/quiz-worksheet-one-to-one-functions.html

https://www.mathworksheets4kids.com/function/identifying/graphing-1.pdf

https://www.statista.com/statistics/1103623/philippines-coronavirus-covid-19-cases-by-residence/

E-SITES

REFERENCES

Acknowledgements Writer: Ariel R. Rogon – MT II Editor: Dr. John Rainier Rizardo, Master Teacher II Reviewer: Remylinda T. Soriano, EPS, Math Angelita Z. Modesto, PSDS

George B. Borromeo, PSDS

Management Team: Maria Magdalena M. Lim- Schools Division Superintendent-Manila,

Aida H. Rondilla-Chief Education

Supervisor

Lucky S. Carpio-EPS and

Lady Hannah C Gillo, Librarian II-LRMS

https://www.youtube.com/watch?v=nE_ykVScQww&t=3shttps://study.com/academy/practice/quiz-worksheet-one-to-one-functions.htmlhttps://www.mathworksheets4kids.com/function/identifying/graphing-1.pdfhttps://www.statista.com/statistics/1103623/philippines-coronavirus-covid-19-cases-by-residence/https://www.statista.com/statistics/1103623/philippines-coronavirus-covid-19-cases-by-residence/

Before starting the module, I want you to set aside other tasks that will disturb you while enjoying the lessons. Read the simple instructions below to successfully enjoy the objectives of this kit. Have fun!3. Perform all the provided activities in the module. Activities - This is a set of activities you will perform with a partner.

Documents

GENERAL MATHEMATICSpauljorelsantos.weebly.com/uploads/9/2/9/2/92921742/... · 2020. 8. 15. · Learning Module for General Mathematics 2 PARTS OF THE MODULE • Expectations - These