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Download In about 48 hours your document will be visible to the other user and more Exercises Mathematics in PDF only on Docsity! a DepED DEPARTMENT Mf OF EDUCATION, General Mathematics Quarter 1 - Module 5: Rational Functions, Equations and Inequalities General Mathematics Alternative Delivery Mode Quarter 1 — Module 5: Rational Functions, Equations and Inequalities First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Development Team of the Module Writer: 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, and Jerome A. Chavez Illustrator: Hanna Lorraine G. Luna Layout Artist: Noel Rey T. Estuita Management Team: Wilfredo E. Cabral, Job S. Zape Jr., Eugenio S. Adrao, Elaine T. Balaogan, Hermogenes M. Panganiiban, Babylyn M. Pambid, Josephine T. Natividad, Anicia J. Villaruel, Dexter M. Valle Printed in the Philippines by Department of Education Region IV-A CALABARZON Office Address: Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800 Telefax: 02-8682-5773/8684-4914/8647-7487 E-mail Address: region4a@deped.gov.ph This module has the following parts and corresponding icons: What I Need to Know What I Know What’s In What’s New What is It What’s More What I Have Learned What I Can Do Assessment Additional Activities Answer Key This will give you an idea of the skills or competencies you are expected to learn in the module. This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module. This is a brief drill or review to help you link the current lesson with the previous one. In this portion, the new lesson will be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity or a situation. This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills. This comprises activities for independent practice to solidify your understanding and skills of the topic. You may check the answers to the exercises using the Answer Key at the end of the module. This questions or blank sentence/paragraph to be filled in to process what you learned from the lesson. includes This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns. This is a task which aims to evaluate your level of mastery in achieving the learning competency. In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also tends retention of learned concepts. This contains answers to all activities in the module. At the end of this module you will also find: References This is a list of all sources used in developing this module. The following are some reminders in using this module: 1. aa Rw Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises. Don’t forget to answer What I Know before moving on to the other activities included in the module. Read the instruction carefully before doing each task. Observe honesty and integrity in doing the tasks and checking your answers. Finish the task at hand before proceeding to the next. Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it! What I Need to Know This module was designed to help learners gain understanding about rational functions. It is composed of two lessons. The first lesson tackles about representing real life situations using rational functions, and the second lesson will delve about distinguishing rational function, rational equation and rational inequality. It is assumed that the learners already grasp full understanding with functions which was found on the previous modules. The first part of this module covers varied situations that can be seen in real life such as budgeting distance and concentration of medicine in the blood while the second lesson will proceed to deeper portion or rational sentences. It is hoped that upon exploring this learning kit you will find the eagerness and enthusiasm in completing the task required. Best of luck! After going through this module, you are expected to: 1. represent real — life situations using rational functions 2. distinguishes rational function, rational equation and rational inequality What I Know Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper. 1. A truck that delivers essentials in remote areas can travel 85 kilometers. Which of the following expresses the velocity v as a function of travel time t in hours? a. v(t)= = b. vQ== c. tw) == d. t= IA c. d. 12.Which of the following is considered rational inequality? a.xt+v3<5 x+5 b. f@) == x+5 ce 5 2 5 d.x42 028 4 13.Which of the following is considered rational equation? a.4+5=9 x24+5 “xt vBx+1 ce. 2=— x+2 d.2=21 x42 14. In the equation: = =x+5,what symbol must be replaced with 5 to make the equation a rational function? ay b. v5 as d.5x (ete 15. What symbol is present in the equation y === for not considering it as rational function? ay b. = Cc. xt2 d. vVx?-+3 Representing Real - life Situations Using Rational Functions Rational functions can model a number of real-life situations. One particular example is the help that is extended by the government to the citizen during the time of pandemic. Majority of our fellow citizens experienced hardship and required help coming from the government. As a response, they provided a particular amount to a certain percentage of the population that can be represented as rational function to determine how much either in cash or kind an individual may receive. However, it is not enough that only the government will take part to solve this crisis everyone can be part of the solution if we played our role properly. Real-life situations that involve rational functions is mostly seen in economics and science however other disciplines also incorporate this concept. If you wonder how rational function can help, you can explore this module. What’s In ADMISSION CARD Listed below are the skills and competencies you should possess before proceeding to this lesson. Read the statements and assess yourself about your level of understanding by answering yes if you agree and no if otherwise. Statement Yes No 1. Ican represent real-life situations using function 3. Ican recognize polynomial functions e If your answer to all the items is yes then you are confident to proceed to the next lesson. e If you answered no to any of the statements there is a need for you to have a quick review on the following: 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. Consider the examples below and reflect if you are confident enough to proceed. 1. Write a function C that represent the cost of buying alcohol a, if an alcohol costs P155.00 C(a) = 155a 2. Acommuter pays P20.00 for a tricycle ride for the first 5 km and an additional P 0.75 for every succeeding distance d in kilometer. Represent the situation as function F(d) = 20, if O<d<5 F(d) = 20 + 0.75(d), if d>5 Let n be a nonnegative integer, and let ay,ay-1,...,@2,01,d9 be real numbers with a, + 0. The function f(x) = a,x" + ay_yx""14+...4+a2x? +a,x +d, is called a polynomial function of x with degree n. The coefficient a, is called the leading coefficient, and ay is the constant. Here are the examples of polynomial functions of particular degree together with their names: Polynomial Degree Special Name f) =3 0 Constant Function f@) = -2x4+1 1 Linear Function f Ce) = Bx? — 5x +2 2 Quadratic Function f(x) = 4x3 + 2x-7 3 Cubic Function At this point you may now proceed to the next section of this module! Notes to the Teacher The teacher may reiterate the formula in finding the distance which is d=rt and the other relationships derived from this formula such r =< and t= £ These relationships will be helpful in meeting the expected outcome of this module. The following are examples of rational functions: 3_. 1. r(x) =f xe 1 functions, denominator has restriction because it should not be equal to zero 2. f(x) =5x #0 degree O, the denominator is a polynomial function and it must not be equal to 0 There are different scenarios or real-world relationships that can be modeled by rational functions, let us take the following examples: 1. The Local Government Unit allotted a budget of P100,000.00 for the feeding program in the Day Care Center. The amount will be divided equally to all the pupils in the Day Care Center. Write an equation showing the relationship of the allotted amount per pupil represented by jfx) versus the total number of children represented by x Showing the relationship in tabular form we will arrived at ; Both numerator and denominator are polynomial ; The numerator 1 is a polynomial function with a No. of children (x) 10 20 50 100 200 Allocated amount per child P10,000 P5000 P2000 P1000 PS00 Notice that as the number of children increase the amount allocated per child decrease. In writing a representation we will arrived at f(x) = 2. Suppose a benefactor wants to supplement the budget allotted for each child by donating additional P650.00 per child. If h(x) represents the new amount allotted 100000 per child, construct a function representing the relationship. Using the table we used earlier: No. of children | 10 (x) 20 50 100 200 Allocated P10,000.00 | P5000.00 | P2000.00 | 1000.00 | P500.00 amount per | +P650.00 +P650.00 +P650.00 +P650.00 | +P650.00 child Thus, the representation of the rational function is g(x) = 200000 + 650 x 3. Acar is to travel a distance of 70 kilometers. Express the velocity (v) as a function of travel time (4 in hours. Let us first show the relationship using a table. Remember that as time increases in travelling the velocity or the speed of a car will decrease Time (hours) 1 2 3 5 10 Velocity 70 35 23.33 14 7 (km/hr) Thus, the function v(t) = 2 can represent v as a function of t What’s More Read each situation carefully to solve each problem. Write your answer on a separate sheet of your paper. Independent Practice 1 School is Cool During the first quarter of the school year the officers —elect of the Supreme Student Government decided to divide their budget evenly to the different committees. If their budget is P35,000 construct a function M which would give the amount of money each of the n number of committees would receive. a. You may construct a table to aid you in determining the relationships between quantities Number of 2 4 6 8 Committees Amount allocated for 10 each committee b. Write the rational function that represents the situation Independent Assessment 1 Pastry Corner Manuel has 10 cups of flour to be used in baking cakes, he wanted to split it evenly among the containers that he will use so that he can adjust the measurements of other ingredients. Construct a function C which would give the number of cups of flour each of the number of containers n will have. Independent Practice 2 Medicine Dosage Let C(t) = = be the function that describes the concentration of a certain medication in the bloodstream overtime t. If 4 hours have passed after the medicine was intake, how concentrated is it in the blood? a. What is the rational function that serves as the model? b. How are you going to determine the concentration of the medicine given the rational function and the number of hours? Independent Assessment 2 How Long The distance between the school and your home is 5 kilometers. Express velocity (v) as a function of travel time (t) in hours 11 Rubrics for the Task Categories Excellent Fair Poor 3 2 1 Budgeting Excellent Some understanding | Little to no understanding in | in creating a plan for | understanding in creating a plan for | spending the money | creating a plan for spending the money spending the money Planning The goal set is | Thegoalsetishardto| The goal set is not achievable and | achieve achievable and not realistic realistic Accuracy of| The computation in| The computation in| There is no attempt Solution obtaining the desired | obtaining the desired |in computing the profit using the profit function is correct profit using the profit function has flaws desired profit using the profit function 14 Lesson . . Rational Functions, Equations and Inequalities It was defined in the previous lesson that rational functions are expressed as a ratio of two polynomials P and Q. The value of rational functions is defined for all real numbers x, except for the value of x that makes the denominator zero. There are different relationships between rational expressions. It may involve inequality, equality and functions and that is what we are going to dig deeper in this lesson What’s In Admission Card Listed below are the skills and competencies you should possess before proceeding to this lesson. Read the statements and assess yourself about your level of understanding by answering yes if you agree and no if otherwise. Statement Yes No 1. Ican describe a rational expression 3. I can distinguish whether an expression is rational or not e Ifyour answer to all the items is yes then you are confident to proceed to the next lesson. e Ifyou answered no to any of the statements, there is a need for you to have a quick review on the following: A rational expression can be described as a ratio or quotient of two polynomials. Let us look at the examples: Consider the following algebraic expressions, determine whether they are rational or not and state the reason. 2H! . . 4 . . 1. — ; Rational expression because it is a ratio of two polynomials 2 = ; Rational expression because 1 and x-5 are polynomials 3. = ; Not a rational expression since the numerator is not a polynomial 4. x45 ; Rational expression because the numerator x+5 and denominator 1 are polynomials 15 7 SO What’s New Read and analyze the advertisement below Wanted!!! Call for Applicants for Contract of Service Workers For the purpose of 2020 census of population and housing, Philippine Statistics Authority is hiring enumerators. Any male or female Age - at least 18 years old and at most 45 years old Educational Attainment — at least SHS graduate Salary - P700.00 per day Physically fit for field work Interested applicants must submit application letter and Personal Data Sheet (PDS) to PSA- Quezon Questions 1. What is the advertisement all about? 2. What job is offered by PSA? 3. Who can apply for the position? 4. How will you translate the required age mathematically? 5. Represent the salary (S) that will be received as a function of number of days used for work (n). 6. What are the mathematical symbols that you used in answering questions numbers 4 and 5? 7. Do you think the age requirement for the job is fair enough or is there a bias? Why do you say so? 8. Why do you think having a census is important? 16 2. Which of the following is an example of rational function? A. 2-3-4 x xt+1 B. vx +2 = f(x) o Baas x xt+1 d.y=2x+3 3. What symbol will be replaced with the equal sign in the equation = = 2x? to make it an inequality? Independent Assessment 2 Read the statements carefully and choose the letter that corresponds to the correct answer. 1. How do you classify = wz 3y? a. Rational Equation b. Rational Function c. Rational Inequality d. None of these 2. Which of the following is an example of rational function? a. 5x3 = xt4 b. V2x? = f(x) x a Coa x+1 d.x+y=2x+3 3. What symbol will be replaced with the "<" sign in = < 2x? to make it a rational equation? 19 What I Have Learned 3-2-1 Action What are the distinct features of rational function, rational equation and rational inequality? 1. Rational Function 2. Rational Equation 3. Rational Inequality What are the differences of the following: 1. Rational Function and Rational Equation 2. Rational Equation and Rational Inequalities Share one tip or suggestion on how to identify whether the given is a rational function, rational equation or rational inequality 1. 20 © ‘@@) i What I Can Do 1G You are analyzing your bills in MERALCO for the consecutive months. Using your bill at home express the amount per kwh (A) as a function of kilowatt per hour (k) consumed for the last two consecutive months. Then create austerity plan for the next month. (austerity- measures taken to reduce spending) Austerity Plan Bill for the last two consecutive months: Create a rational inequality showing the relationship between the bill in the last consecutive months: List down the austerity measures to be undertaken to reduce the electricity consumption: 21 6. Which of the following is considered rational inequality a. V5<5 x+15 b. = 3 2x+15 ce. 82—— 14 d. x42028 4 7. Which of the following is considered rational equation? a, Sx+ 8 2425 b. x+5 v3x+1 x2 _ 4d © 2x42 8. In the equation: = =x+8,what symbol must be replaced with 8 to make the equation rational function? ay b. v5 cs d. . Sx V3x?2+3 for not considering it as 2x42 9. What symbol is present in the equation y= rational function? ay b. = Cc. Qx+2 d. v3x7 +3 10. A delivery track that will bring cargo will travel 80 kilometers. Which of the following expresses the velocity v as a function of travel time t in hours? a. v(t)= = b. v(t) == Cc. tv) =2 d. tw)=4 11.If the truck in problem no. 1 was delayed by 4 hours due to the checkpoints that it passed through what will be the time tas a function of velocity v in km/hr? a. v(t)= a4 b. v(t) = 544 ce. tv)= a4 d. tw)=2+4 24 12.As a cure to the epidemic that spread in the whole country the Department of Health (DOH) released a new drug that is subject for experimentation, supposed that c(t) = Hin mg/mL)represents the concentration of a drug in a patient’s blood stream in t hours, how concentrated is the drug after 4 hours of administration? a. 4.67mg/mL b. 3.33mg/mL c, 2.67mg/mL d. 1.33mg/mL 13.If the distance from Manila to Batangas is approximately 109 kilometers, which of the following pertains to the function (s), where s is the speed of travel that describes the time it takes to drive from Manila to Batangas? 109 a. t(s)= > b. t(s)= S s()= d. s(t) =a For numbers 14-15 use the problem below: Due to the Enhanced Community Quarantine, Toy’s for Her and Him temporarily stopped its operation and to help the employees the owner decided to split evenly its total revenue of P45, 000.00 14.If the number of employees is represented by x , which function represents the amount each received? a. f(x) = 45000x b. f(x) =x + 45000 c. f= aso0n d. f(x) =x — 45000 15.If the owner held a fund-raising activity that aimed to help her employees and collected P1500.00 per employee, which of the following represents the total amount an employee will receive? a. f(x) = 45000x + 1500 b. f(x) =x + 45000 + 1500 c. fx) = 2% 41500 x d. f(x) = x— 45000 + 1500 25 % \ additional Activities RS To strengthen your skills in determining rational functions, rational equations and rational inequality, construct five examples of each category. Rational Equation Rational Function Rational Inequality 1. 1. 1. 2 2 2 3 3 3 4. 4. 4. 5 5 5 26