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CO_Q1_General Mathematics SHS

General Mathematics

Quarter 1 – Module 4:

Solving Real- Life Problems

Involving Functions

General Mathematics Alternative Delivery Mode Quarter 1 – Module 4: Solving Real- Life Problems Involving Functions First Edition, 2021

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Printed in the Philippines by ________________________

Department of Education – Region 4-A CALABARZON

Office Address: Gate 2 Karangalan Village, Brgy. San Isidro, Cainta, Rizal Telefax: 02-8682-5773/8684-4914/8647- E-mail Address: [email protected]

Development Team of the Module Writers: Rey Mark R. Queaño and Ann Michelle M. Jolo Editors: Elizabeth B. Dizon, Anicia J. Villaruel and Roy O. Natividad Reviewers : Fritz A. Caturay, Necitas F. Constante, Dexter M. Valle, Jerome A. Chavez, Azenith G. Mercado and Ceclia A. Ong Illustrator: Dianne C. Jupiter Layout Artist: Noel Rey T. Estuita and Sayre M. Dialola Management Team: Francis Cesar B. Bringas Job S. Zape, Jr. Ramonito Elumbaring Reicon C. Condes Elaine T. Balaogan Fe M. Ong-ongowan Hermogenes M. Panganiban Phillip B. Gallendez Josephine T. Natividad Anicia J. Villaruel Dexter M. Valle

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by- step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

CO_Q1_General Mathematics SHS

What I Need to Know

This module was designed and written to help you solve problems involving functions bearing in mind that you already know how to represent real – life situation using functions including piece-wise functions, evaluate functions and perform operations on functions. These skills will aid you in attaining success on this module.

Solving problems involving functions is essential in predicting values that will help in decision making process. This module covers varied situations that can be seen in real life such as travel fares, monthly bills sales and the like. It is hoped that upon exploring this learning kit you will find the eager and enthusiasm in completing the task required. Best of luck!

After going through this module, you are expected to:

1. represent situations as functions and evaluate functions to determine the required quantity
2. apply concepts learned in solving real-life problems involving functions
3. solve problems involving functions

CO_Q1_General Mathematics SHS

For numbers 6 – 7 use the problem below:

Cotta National High School has 1,200 students enrolled in 2016 and 1,500 students in 2019. The student population P grows as a linear function of time (t) , where t is the number of years after 2016.

6. Which of the following functions represents the student population that relates to time t? a. P (t )= 100t + 1, b. P (t) = 1,200t - c. P( t) = 1,200t + 100 d. P (t ) = 100t – 1,
7. How many students will be enrolled in Cotta National High School by 2020? a. 1, b. 1, c. 1, d. 1,

For numbers 8 – 10 use the problem below:

A proposed tricycle fare would charge ₱20.00 for the first 5 km of travel and ₱0.75 for each additional kilometer over the proposed fare.

8. Find the fare function f(x) where x represents the number of kilometers travelled. a. f(x) = 20 + 0.75x b. f(x) = 20 - 0.75x c. f(x) =16.25 + 0.75x d. f(x) = 6.25 - 0.75x
9. How much is the proposed fare for distance of 3 km? a. ₱4. b. ₱8. c. ₱12. d. ₱20.
10. Find the proposed fare for distance of 55 km. a. ₱57. b. ₱91. c. ₱60. d. ₱105.

CO_Q1_General Mathematics SHS

11. The cost of producing x tools by a B&G Corp. is given by C(x) =₱1,200.00 (x) + ₱5,500.00. How much is 100 tools? a. ₱6,700. b. ₱12,550. c. ₱125,500. d. ₱551,200.

For numbers 12 - 15 use the problem below:

Mark charges ₱100.00 for an encoding work. In addition, he charges ₱5.00 per page of printed output.

12. Find a function f (x) where x represents the number of pages of printed out. a. f(x) = 100 + 5 x b. f(x) = 100 - 5 x c. f(x) = 100x + 5 d. f(x) = 100x – 5
13. How much will Mark charge for 55-page encoding and printing work? a. ₱275. b. ₱175. c. ₱375. d. ₱475.
14. How many pages were printed if Mark received a payment of ₱600.00? a. 100 pages b. 80 pages c. 60 pages d. 50 pages
15. If Mark offers a promo to loyal costumer that the first 20 pages of the printed output will be free of charge, how much will he charge to a loyal customer who printed 70 pages of output? a. ₱250. b. ₱50. c. ₱350. d. ₱450.

CO_Q1_General Mathematics SHS

1. Being able to understand a problem presented is the first thing that we learn on how to solve problems in Mathematics. Careful reading leads you to thorough analysis in the identification of given facts and in determining the required or the unknown quantity. Precise label of the known and unknown quantities will help you set up a direction towards the solution.
2. A function is a rule of correspondence between two non-empty sets, such that to each element of the first set called domain, there corresponds one and only one element of the second set called range.
3. Functions are used to model real-life situations and in representing real-life situations the quantity of one variable depends or corresponds to or mapped onto another quantity.
4. Piece-wise function are functions that may be represented by a combination or of equations.
5. If a function f is defined by y = f(x) and an independent variable x is found by substituting x into the function rule then it undergoes a process of evaluating function. Moreover, you also studied fundamental operation can be applied to two or more functions to form a new function. Such operations are addition, subtraction, multiplication and division.

Consider the examples below and reflect if you are confident enough to proceed

1. Write a function C that represent the cost of buying x facemask, if a facemask cost ₱65. C(x) = 65x
2. A commuter pays ₱ 9.00 for a jeepney fare for the first 5 km and an additional ₱ 0.75 for every succeeding distance d in kilometer. Represent the situation as function F(d) = 9, if 0<d< F(d) = 9 + 0.75(d) , if d>
3. If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 6, evaluate: a. f(4) b. f(-2) c. f(-x)

Solution : a. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 6 b. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 6 c. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 6 f(4) = 4+ 6 𝑓𝑓(−2) = (−2) + 6 𝑓𝑓(−𝑥𝑥) = −𝑥𝑥 + 6 f(4) = 10 𝑓𝑓(−2) = 4

CO_Q1_General Mathematics SHS

4.Le 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 3t and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2. Find a. 𝑓𝑓(3) + 𝑔𝑔(−2) b. 𝑓𝑓(4) − 𝑔𝑔(0)

c. 𝑓𝑓(𝑥𝑥) ∙ 𝑔𝑔(𝑥𝑥) d. 𝑓𝑓 𝑔𝑔((^98 )) Solution: a. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 3 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2 𝑓𝑓(3) = 3 + 3 𝑔𝑔(−2) = (−2) − 2 𝑓𝑓(3) = 6 𝑔𝑔(−2) = − 4 𝑓𝑓(3) + 𝑔𝑔(−2) = 6 + (−4) 𝑓𝑓(3) + 𝑔𝑔(−2) = 2

b. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 3 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2 𝑓𝑓(4) = 4 + 3 𝑔𝑔(0) = 0 − 2 𝑓𝑓(4) = 7 𝑔𝑔(0) = − 2 𝑓𝑓(4) − 𝑔𝑔(0) = 7 − (−2) 𝑓𝑓(4) − 𝑔𝑔(0) = 9

c. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 3 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2 𝑓𝑓(𝑥𝑥)^ ∙ 𝑔𝑔(𝑥𝑥) = (𝑥𝑥 + 3)(𝑥𝑥 − 2) 𝑓𝑓(𝑥𝑥)^ ∙ 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 2 + 𝑥𝑥 − 6

d. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 3 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2 𝑓𝑓(9) = 9 + 3 𝑔𝑔(8) = 8 − 2 𝑓𝑓(9) = 12 𝑔𝑔(8) = 6 𝑓𝑓(9) 𝑔𝑔(8)

At this point you may now proceed to the next section of this module!

Notes to the Teacher

The teacher may also point out the importance of the concept of zero of linear function in solving problems involving functions. The zero of a linear function f(x) is the real number a such that f(a)=0. This suggest that the zero of linear function is found by equating it to zero and then solving the resulting equation for x. This will be used in the latter example in the module.

CO_Q1_General Mathematics SHS

What is It

Decision making is always part of our lives, from the moment we wake-up we start to decide the proper action to undertake be it minor or major decisions. In the problem presented, Miguel is about to consider the jeepney and tricycle fare in making decision. He will only pay ₱17.25 in a jeepney while ₱24.00 in a tricycle. With this, it will be more economical if he will choose to ride in a jeepney. However, the cost of the fare is just one of the factors. There are times that convenience is also considered in choosing the mode of transportation. Aside from not being crowded and you can reach your destination faster. Therefore, in deciding the mode of transportation the priority of the commuters whether to be more economical or to meet convenience is considered.

In the previous problem we determine the cost of the fare by using a table wherein we repeatedly add the fare charge per kilometer. Thus, this type of problem can be solved using functions, and at this point let us determine how we are going to do that.

Example no. 1

LET’S TRAVEL

A proposed Light Rail Transit System Line 1 (LRT-1) fare would charge ₱18.00 for the first four stations and ₱5.00 for each additional station over the proposed fare.

a. Find the fare function f(x) where x represents the number of stations traveled b. Find the proposed fare for 15 stations c. Find the proposed fare for 20 stations

To solve problems that involve functions you can employ George Polya’s 4-step rule.

George Polya’s 4 – Step Rule

1. Explore. This step involves careful reading, analyzing, identifying the given and unknown facts in the problem and expressing the unknown in terms of variables.
2. Plan. In this step writing an equation that describes the relationships between or among the variables is involve.
3. Solve. This step requires working out with the written equation and other number relations to determine the required quantities that answer the question in the problem.
4. Check. The final step that employ the use of other approaches to examine the appropriateness of the answer.

CO_Q1_General Mathematics SHS

Solution

a. Explore. Since the first step involves analysis and proper labeling of the known and unknown facts we will let x = number of stations traveled. There are also some conditions that was set in the problem such as the cost of fare which is set up to 4 stations only thus we can represent x – 4 = number of stations traveled over and above 4 stations

Plan. In writing an equation that will represent the relationship between the known and unknown quantities, since we know that if we travelled up to 4 stations we must pay P18, we can represent it as f(x) = 18 for 0 < 𝑥𝑥 ≤ 4

However, if we travelled more than 4 stations the cost of the fare have different method of computation so we need to consider it. Since the cost of every station after the 4th^ station is ₱5.00 we will now obtain f(x) = 18 + 5( x – 4)

Now simplifying the equation will lead us to: f(x) = 18 + 5 x – 20 f(x) = 5 x – 2

At this point we can say that the fare function is f(x) = 5x - 2

b. Solve. To find the fare charge for 15 stations the fare function f(x) = 5 x - will be used and 15 will be substituted to the function f(15) = 5(15) – 2 = 73 By evaluating the function we obtained f(15) = 73

Check. To check whether we arrived at the correct solution you can use table or graph. Thus. the proposed fare for 15-station travel is ₱73.

c. f(20) = 5 (20) – 2 = 98

The proposed fare for 20 – station travel is ₱ 98

CO_Q1_General Mathematics SHS

What’s More

Read each situation carefully to solve each problem. Write your answer on a separate sheet of your paper.

Independent Practice 1

a. Find the profit function P(x) where x represents the number of cookies sold Hint: Profit = Total Revenue – Total Cost Total Revenue = Price per unit x quantity sold Total Cost = Total variable cost + fixed cost ____________________________________________________________

b. If 146 cookies were sold, how much is the total profit?

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c. How many cookies must be made and sold to break even? Hint: Break even point is the zero of P(x)

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d. How many cookie should be sold to gain a profit of ₱250.00?

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Business As Usual

Bakers’ Club is trying to raise funds by selling premium chocolate chip cookies in a school fair. The variable cost to make each cookie is ₱15.00 and it is being sold for ₱25.00 So far the organization has already shelled out ₱790.00 for the cookie sale

CO_Q1_General Mathematics SHS

Independent Assessment 1

Independent Practice 2

a. Find the monthly cost function C(x) where x represents the number of minutes used. Hint: Monthly Cost = Plan Cost + Additional Charge per Minute ________________________________________________________________

b. How much is the monthly cost incurred if the owner used 180 minutes of call?____________________________________________________________

c. How much is the monthly cost of the plan if the owner used 300 minutes of call?______________________________________________________________

Baker’s Nook Elisha just opened a bakery along Macalintal Avenue which sells fresh doughnuts. The selling price is ₱20.00 per doughnut and the cost of making it is ₱8.00 The daily operating expense is ₱600.00. a. Find the profit function P (x) where x represent the number of doughnuts sold. b. If 100 doughnuts were sold, what is the total profit? c. How many doughnuts must be made and sold to break even? d. How many doughnuts should be sold to gain a profit of ₱600.00?

Hello!

A certain cellphone company offers a plan that costs ₱1,200.00 per month. The plan includes 180 free minutes of call and charges ₱7. for each additional minute of usage.

CO_Q1_General Mathematics SHS

What I Can Do

You wanted to join a booth fair, and you are aiming to get a profit that is twice as your capital. Your starting capital is ₱15,000.00. Make a financial plan for the booth that you will set up and the product that you will sell. You may use the sample plan below:

FINANCIAL PLAN

Product: _________________________

Description of product: ________________________ Goal: _____________________________________

Capital: ₱15,000.

Fixed Cost (Labor, Machineries, Expenses for the booth etc): _______

Variable Cost (Materials, Ingredients, etc): ____________

Profit function: ___________________

Prove that profit function will yield an amount that is twice the capital

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CO_Q1_General Mathematics SHS

Rubrics for the Task

Categories Excellent 3

Fair 2

Poor 1

Budgeting Excellent understanding in creating a plan for spending the money

Some understanding in creating a plan for spending the money

Little to no understanding in creating a plan for spending the money

Planning The goal set is achievable and realistic

The goal set is hard to achieve

The goal set is not achievable and not realistic

Accuracy of Solution

The computation in obtaining the desired profit using the profit function is correct

The computation in obtaining the desired profit using the profit function has flaws

There is no attempt in computing the desired profit using the profit function