General Mathematics Learner's Material Department of Education Republic of the Philippines PDF

Title General Mathematics Learner's Material Department of Education Republic of the Philippines
Author Claire Hisman
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Summary

General Mathematics PY Learner’s Material O C D This learning resource was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or E universities. We encourage teachers and other education stakeholders to email their feedback, comments and recommendation...


Description

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General Mathematics

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Learner’s Material

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This learning resource was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments and recommendations to the Department of Education at [email protected].

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We value your feedback and recommendations.

Department of Education Republic of the Philippines

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

General Mathematics Learner’s Material First Edition 2016 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.

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Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this learning resource are owned by their respective copyright holders. DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seeking permission to use these materials from their respective copyright owners. All means have been exhausted in seeking permission to use these materials. The publisher and authors do not represent nor claim ownership over them.

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Only institutions and companies which have entered an agreement with FILCOLS and only within the agreed framework may copy from this Learner’s Material. Those who have not entered in an agreement with FILCOLS must, if they wish to copy, contact the publishers and authors directly. Authors and publishers may email or contact FILCOLS at [email protected] or (02) 435-5258, respectively.

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Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD

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Development Team of the General Mathematics Learner’s Material Dr. Debbie Marie B. Verzosa Francis Nelson M. Infante Paolo Luis Apolinario Jose Lorenzo M. Sin Regina M. Tresvalles Len Patrick Dominic M. Garces Reviewers Leo Andrei A. Crisologo Lester C. Hao Shirlee R. Ocampo Eden Delight P. Miro Jude Buot Eleanor Villanueva Cover Art Illustrator Quincy D. Gonzales

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Layout Artist Mary Ann A. Mindaña

Management Team of the General Mathematics Learner’s Material Bureau of Curriculum Development Bureau of Learning Resources

Printed in the Philippines by Lexicon Press Inc. Department of Education-Bureau of Learning Resources (DepEd-BLR) Office Address: Ground Floor Bonifacio Building, DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (02) 634-1054 or 634-1072 E-mail Address: [email protected]

ii All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

TABLE OF CONTENTS I. Functions Lesson 1: Functions............................................................................................. 1 Lesson 2: Evaluating Functions ......................................................................... 10 Lesson 3: Operations on Functions ................................................................... 13

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II. Rational Functions Lesson 4: Representing Real-Life Situations Using Rational Functions ............. 21 Lesson 5: Rational Functions, Equations, and Inequalities ................................ 24 Lesson 6: Solving Rational Equations and Inequalities ...................................... 25 Lesson 7: Representations of Rational Functions .............................................. 35 Lesson 8: Graphing Rational Functions ............................................................. 44

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III. One-to-One Functions Lesson 9: One-to-One functions ........................................................................ 60 Lesson 10: Inverse of One-to-One Functions..................................................... 62 Lesson 11: Graphs of Inverse Functions............................................................ 67

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IV. Exponential Functions Lesson 12: Representing Real-Life Situations Using Exponential Functions ...... 77 Lesson 13: Exponential Functions, Equations, and Inequalities ......................... 82 Lesson 14: Solving Exponential Equations and Inequalities............................... 83 Lesson 15: Graphing Exponential Functions ...................................................... 88 Lesson 16: Graphing Transformations of Exponential Functions ....................... 92 Logarithmic Functions Lesson 17: Introduction to Logarithms ............................................................... 99 Lesson 18: Logarithmic Functions, Equations, and Inequalities ....................... 103 Lesson 19: Basic Properties of Logarithms ...................................................... 104 Lesson 20: Laws of Logarithms ....................................................................... 106 Lesson 21: Solving Logarithmic Equations and Inequalities ............................. 111 Lesson 22: Graphing Logarithmic Functions .................................................... 124 Lesson 23: Illustrating Simple and Compound Interest ................................... 135

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V.

iii All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

VI. Simple and Compound Interest Lesson 24: Simple Interest .............................................................................. 137 Lesson 25: Compound Interest ....................................................................... 144 Lesson 26: Compounding More than Once a Year .......................................... 150 Lesson 27: Finding Interest Rate and Time in Compound Interest .................. 158

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VII. Annuities: Lesson 28: Simple Annuity .............................................................................. 168 Lesson 29: General Annuity ............................................................................ 183 Lesson 30: Deferred Annuity ........................................................................... 199

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VIII. Basic Concepts of Stocks and Bonds Lesson 31: Stocks and Bonds ......................................................................... 208 Lesson 32: Market Indices for Stocks and Bonds ............................................ 217 Lesson 33: Theory of Efficient Markets............................................................ 222

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Logic Lesson 36: Propositions .................................................................................. 240 Lesson 37: Logical Operators.......................................................................... 246 Lesson 38: Truth Tables .................................................................................. 257 Lesson 39: Logical Equivalence and Conditional Propositions ........................ 263 Lesson 40: Valid Arguments and Fallacies ...................................................... 270 Lesson 41: Methods of Proof ........................................................................... 283

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IX. Basic Concepts of Loans Lesson 34: Business and Consumer Loans .................................................... 225 Lesson 35: Solving Problems on Business and Consumer Loans .................. 226

iv All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

Lesson 1: Functions Learning Outcome(s): At the end of the lesson, the learner is able to represent reallife situations using functions, including piecewise functions. Lesson Outline: 1. Functions and Relations 2. Vertical Line Test

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3. Representing real-life situations using functions, including piecewise functions.

Definition: 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 relation is a set of ordered pairs (x,y).

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Definition: A function is a relation where each element in the domain is related to only one value in the range by some rule.

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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. Using functional notation, we can write f(x) = y, read as “f of x is equal to y.” In particular, if (1, 2) is an ordered pair associated with the function f, then we say that f(2) = 1.

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Example 1. Which of the following relations are functions?

Solution.

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The relations f and h are functions because no two ordered pairs have the same x-value but different y-values. Meanwhile, g is not a function because (1,3) and (1,4) are ordered pairs with the same x-value but different y-values. Relations and functions can be represented by mapping diagrams where the elements of the domain are mapped to the elements of the range using arrows. In this case, the relation or function is represented by the set of all the connections represented by the arrows.

1 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

Example 2. Which of the following mapping diagrams represent functions?

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Solution.

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X

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The relations f and g are functions because each value y in Y is unique for a specific value of x. The relation h is not a function because there is at least one element in X for which there is more than one corresponding y-value. For example, x=7 corresponds to y = 11 or 13. Similarly, x=2 corresponds to both y=17 or 19.

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A relation between two sets of numbers can be illustrated by a graph in the Cartesian plane, and that a function passes the vertical line test.

The Vertical Line Test A graph represents a function if and only if each vertical line intersects the graph at most once.

2 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

Example

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Which

of

the

following

can

be

graphs

of

functions?

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a.)

b.)

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D

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e.)

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c.)

d.) Solution. Graphs a.), b.), c.) are graphs of functions while d.) and e.) are not because they do not pass the vertical line test. 3 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

Important Concepts.  

Relations are rules that relate two values, one from a set of inputs and the second from the set of outputs. Functions are rules that relate only one value from the set of outputs to a value from the set of inputs.

Definition: The domain of a relation is the set of all possible values that the variable x can take. Example 4. Identify the domain for each relation using set builder notation.

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(a) (b) (c) (d) (e) where

is the greatest integer function.

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(f)

Solution. The domains for the relations are as follows: (d) (e) (f)

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(a) (b) (c)

Functions as representations of real-life situations.

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Functions can often be used to model real situations. Identifying an appropriate functional model will lead to a better understanding of various phenomena. Example 5. Give a function C that can represent the cost of buying x meals, if one meal costs P40. Solution. Since each meal costs P40, then the cost function is C(x) = 40x.

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Example 6. One hundred meters of fencing is available to enclose a rectangular area next to a river (see figure). Give a function A that can represent the area that can be enclosed, in terms of x. RIVER y x Solution. The area of the rectangular enclosure is A = xy. We will write this as a function of x. Since only 100 m of fencing is available, then x + 2y = 100 or y = (100 – x)/2 = 50 – 0.5x. Thus, A(x) = x(50 – 0.5x) = 50x – 0.5x2. 4 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

Piecewise functions. Some situations can only be described by more than one formula, depending on the value of the independent variable.

Example 7. A user is charged P300 monthly for a particular mobile plan, which includes 100 free text messages. Messages in excess of 100 are charged P1 each. Represent the monthly cost for text messaging using the function t(m), where m is the number of messages sent in a month.

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Solution. The cost of text messaging can be expressed by the piecewise function:

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Example 8. A jeepney ride costs P8.00 for the first 4 kilometers, and each additional integer kilometer adds P1.50 to the fare. Use a piecewise function to represent the jeepney fare in terms of the distance (d) in kilometers.

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Solution. The input value is distance and the output is the cost of the jeepney fare. If F(d) represents the fare as a function of distance, the function can be represented as follows:

Note that is the floor function applied to d. The floor function gives the largest integer less than or equal to d, e.g. .

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Example 9. Water can exist in three states: solid ice, liquid water, and gaseous water vapor. As ice is heated, its temperature rises until it hits the melting point of 0°C and stays constant until the ice melts. The temperature then rises until it hits the boiling point of 100°C and stays constant until the water evaporates. When the water is in a gaseous state, its temperature can rise above 100°C (This is why steam can cause third degree burns!). A solid block of ice is at -25°C and heat is added until it completely turns into water vapor. Sketch the graph of the function representing the temperature of water as a function of the amount of heat added in Joules given the following information:  The ice reaches 0°C after applying 940 J.  The ice completely melts into liquid water after applying a total of 6,950 J. 5

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

 The water starts to boil (100°C) after a total of 14,470 J.  The water completely evaporates into steam after a total of 55,260 J. Assume that rising temperature is linear. Explain why this is a piecewise function.

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Solution. Let T(x) represent the temperature of the water in degrees Celsius as a function of cumulative heat added in Joules. The function T(x) can be graphed as follows:

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This is a piecewise function because the temperature rise can be expressed as a linear function with positive slope until the temperature hits 0°C, then it becomes a constant function until the total heat reaches 6,950K J. It then becomes linear again until the temperature reaches 100°C, and becomes a constant function again until the total heat reaches 55,260 J.

Solved Examples

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1. Is the relation {(0,0), (1,1), (2,4), (3,9), … (

), …} a function?

Solution.

Yes, it is a function.

6 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

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2. Which of the following diagram represents a relation that is NOT a function?

Solution.

C. All diagrams, except for C, represent a function.

Solution.

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3. Can the graph of a circle be considered a function?

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No, it cannot. A circle will fail the vertical line test.

4. Give the domain of

using set builder notation.

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Solution.

5. Contaminated water is subjected to a cleaning process. The concentration of pollutants is initially 10 mg per liter of water. If the cleaning process can reduce the pollutant by 5% each hour, define a function that can represent the concentration of pollutants in the water in terms of the number of hours that the cleaning process has taken place.

Solution. After 1 hour, the concentration of pollutants is (10)*(0.95). After 2 hours, it is this value, times 0.95, or [(10)*(0.95)](0.95) = 10(0.95) 2. In general, after t hours, the concentration is C(t) = (10)(0.95)t mg per liter of water. 7 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.

6. Squares of side x are cut from each corner of an 8 in x 5 in rectangle (see figure), so that its sides can be folded to make a box with no top. Define a function in terms of x that can represent the volume of this box. 8 inches x

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5 inches

Solution. The length and width of the box are 8 – 2x and 5 – 2x, respectively. Its height is x. Thus, the volume of the box can be represented by the function

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V(x) = (8 – 2x)(5 – 2x)x = 40x – 26x2 + 4x3.

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Solution.

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7. A certain chocolate bar costs P35.00 per piece. However, if you buy more than 10 pieces, they will be marked down to a price of P32.00 per piece. Use a piecewise function to represent the cost in terms of the number of chocolate bars bought.

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8. A school’s fair committee wants to sell t-shirts for their school fair. They found a supplier that sells t-shirts at a price of P175.00 a piece but can charge P15,000 for a bulk order of 100 shirts and P125.00 for each excess t-shirt after that. Use a piecewise function to represent the cost in terms of the number of tshirts purchased. Solution.


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