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General Mathematics Learner’s Material Department of Education Republic of the Philippines 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]. We value your feedback and recommendations. 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. ii 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. 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. 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. Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD 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] 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 Layout Artist Cover Art Illustrator Mary Ann A. Mindaña Quincy D. Gonzales Management Team of the General Mathematics Learner’s Material Bureau of Curriculum Development Bureau of Learning Resources 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. iii TABLE OF CONTENTS I. Functions Lesson 1: Functions ............................................................................................. 1 Lesson 2: Evaluating Functions ......................................................................... 10 Lesson 3: Operations on Functions ................................................................... 13 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 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 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 V. 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 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. iv 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 VII. Annuities: Lesson 28: Simple Annuity .............................................................................. 168 Lesson 29: General Annuity ............................................................................ 183 Lesson 30: Deferred Annuity ........................................................................... 199 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 IX. Basic Concepts of Loans Lesson 34: Business and Consumer Loans .................................................... 225 Lesson 35: Solving Problems on Business and Consumer Loans .................. 226 X. 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 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.
