General Mathematics-Senior Highschool Teaching Guide

TEACHING GUIDE FOR SENIOR HIGH SCHOOL

General Mathematics CORE SUBJECT

This Teaching Guide was collaboratively developed and reviewed by educators from public and private schools, colleges, and universities. We encourage teachers and other education

stakeholders to email their feedback, comments, and recommendations to the Commission on Higher Education, K to 12 Transition Program Management Unit - Senior High School

Support Team at [email protected]. We value your feedback and recommendations.

Commission on Higher Education in collaboration with the Philippine Normal University

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Printed in the Philippines by EC-TEC Commercial, No. 32 St. Louis Compound 7, Baesa, Quezon City, [email protected]

Published by the Commission on Higher Education, 2016 Chairperson: Patricia B. Licuanan, Ph.D.

Commission on Higher Education K to 12 Transition Program Management Unit Office Address: 4th Floor, Commission on Higher Education, C.P. Garcia Ave., Diliman, Quezon City Telefax: (02) 441-1143 / E-mail Address: [email protected]

DEVELOPMENT TEAM

Team Leader: Debbie Marie B. Verzosa, Ph.D.

Writers: Leo Andrei A. Crisologo, Lester C. Hao, Eden Delight P. Miro, Ph.D., Shirlee R. Ocampo, Ph.D., Emellie G. Palomo, Ph.D., Regina M. Tresvalles, Ph.D.

Technical Editors: Mark L. Loyola, Ph.D., Christian Chan O. Shio, Ph.D.

Copy Reader: Sheena I. Fe

Typesetters: Juan Carlo F. Mallari, Regina Paz S. Onglao

Illustrator: Ma. Daniella Louise F. Borrero

Cover Artists: Paolo Kurtis N. Tan, Renan U. Ortiz

CONSULTANTS THIS PROJECT WAS DEVELOPED WITH THE PHILIPPINE NORMAL UNIVERSITY.University President: Ester B. Ogena, Ph.D. VP for Academics: Ma. Antoinette C. Montealegre, Ph.D. VP for University Relations & Advancement: Rosemarievic V. Diaz, Ph.D.

Ma. Cynthia Rose B. Bautista, Ph.D., CHEDBienvenido F. Nebres, S.J., Ph.D., Ateneo de Manila University Carmela C. Oracion, Ph.D., Ateneo de Manila University Minella C. Alarcon, Ph.D., CHEDGareth Price, Sheffield Hallam University Stuart Bevins, Ph.D., Sheffield Hallam University

SENIOR HIGH SCHOOL SUPPORT TEAM CHED K TO 12 TRANSITION PROGRAM MANAGEMENT UNIT Program Director: Karol Mark R. Yee

Lead for Senior High School Support: Gerson M. Abesamis

Lead for Policy Advocacy and Communications: Averill M. Pizarro

Course Development Officers: Danie Son D. Gonzalvo, John Carlo P. Fernando

Teacher Training Officers: Ma. Theresa C. Carlos, Mylene E. Dones

Monitoring and Evaluation Officer: Robert Adrian N. Daulat

Administrative Officers: Ma. Leana Paula B. Bato, Kevin Ross D. Nera, Allison A. Danao, Ayhen Loisse B. Dalena

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

DepEd General Mathematics Curriculum Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter 1 Functions

Lesson 1: Functions as Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Lesson 2: Evaluating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Lesson 3: Operations on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 2 Rational Functions

Lesson 4: Representing Real-Life Situations Using Rational Functions . . . . . . . . . . . . . . 23

Lesson 5: Rational Functions, Equations, and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 28

Lesson 6: Solving Rational Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Lesson 7: Representations of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Lesson 8: Graphing Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter 3 One-to-One and Inverse Functions

Lesson 9: One-to-One Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Lesson 10: Inverse of One-to-One Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Lesson 11: Graphs of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 4 Exponential Functions

Lesson 12: Representing Real-Life Situations Using Exponential Functions . . . . . . . . . . 88

Lesson 13: Exponential Functions, Equations, and Inequalities . . . . . . . . . . . . . . . . . . . . 94

Lesson 14: Solving Exponential Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96

Lesson 15: Graphing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Lesson 16: Graphing Transformations of Exponential Functions . . . . . . . . . . . . . . . . . . . 107

Chapter 5 Logarithmic Functions

Lesson 17: Introduction to Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Lesson 18: Logarithmic Functions, Equations, and Inequalities . . . . . . . . . . . . . . . . . . . . 125

Lesson 19: Basic Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Lesson 20: Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Lesson 21: Solving Logarithmic Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . 136

Lesson 22: The Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Chapter 6 Simple and Compound Interest

Lesson 23: Illustrating Simple and Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Lesson 24: Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Lesson 25: Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Lesson 26: Compounding More than Once a Year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Lesson 27: Finding Interest Rate and Time in Compound Interest . . . . . . . . . . . . . . . . . 185

Chapter 7 Annuities

Lesson 28: Simple Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Lesson 29: General Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Lesson 30: Deferred Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Chapter 8 Basic Concepts of Stocks and Bonds

Lesson 31: Stocks and Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Lesson 32: Market Indices for Stocks and Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Lesson 33: Theory of Efficient Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Chapter 9 Basic Concepts of Loans

Lesson 34: Business Loans and Consumer Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Lesson 35: Solving Problems on Business and Consumer Loans (Amortization and Mortgage) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Chapter 10 Logic

Lesson 36: Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Lesson 37: Logical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

Lesson 38: Constructing Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Lesson 39: Logical Equivalence and Forms of Conditional Propositions . . . . . . . . . . . . . . 285

Lesson 40: Valid Arguments and Fallacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Lesson 41: Methods of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

i

IntroductionAs the Commission supports DepEds implementation of Senior High School (SHS), it upholds the vision and mission of the K to 12 program, stated in Section 2 of Republic Act 10533, or the Enhanced Basic Education Act of 2013, that every graduate of basic education be an empowered individual, through a program rooted on...the competence to engage in work and be productive, the ability to coexist in fruitful harmony with local and global communities, the capability to engage in creative and critical thinking, and the capacity and willingness to transform others and oneself.

To accomplish this, the Commission partnered with the Philippine Normal University (PNU), the National Center for Teacher Education, to develop Teaching Guides for Courses of SHS. Together with PNU, this Teaching Guide was studied and reviewed by education and pedagogy experts, and was enhanced with appropriate methodologies and strategies.

Furthermore, the Commission believes that teachers are the most important partners in attaining this goal. Incorporated in this Teaching Guide is a framework that will guide them in creating lessons and assessment tools, support them in facilitating activities and questions, and assist them towards deeper content areas and competencies. Thus, the introduction of the SHS for SHS Framework.

The SHS for SHS Framework The SHS for SHS Framework, which stands for Saysay-Husay-Sarili for Senior High School, is at the core of this book. The lessons, which combine high-quality content with flexible elements to accommodate diversity of teachers and environments, promote these three fundamental concepts:

SAYSAY: MEANING Why is this important?

Through this Teaching Guide, teachers will be able to facilitate an understanding of the value of the lessons, for each learner to fully engage in the content on both the cognitive and affective levels.

HUSAY: MASTERY How will I deeply understand this?

Given that developing mastery goes beyond memorization, teachers should also aim for deep understanding of the subject matter where they lead learners to analyze and synthesize knowledge.

SARILI: OWNERSHIP What can I do with this?

When teachers empower learners to take ownership of their learning, they develop independence and self-direction, learning about both the subject matter and themselves.

ii

The Parts of the Teaching Guide This Teaching Guide is mapped and aligned to the DepEd SHS Curriculum, designed to be highly usable for teachers. It contains classroom activities and pedagogical notes, and integrated with innovative pedagogies. All of these elements are presented in the following parts:

1. INTRODUCTION Highlight key concepts and identify the essential questions

Show the big picture

Connect and/or review prerequisite knowledge

Clearly communicate learning competencies and objectives

Motivate through applications and connections to real-life

2. MOTIVATION Give local examples and applications

Engage in a game or movement activity

Provide a hands-on/laboratory activity

Connect to a real-life problem

3. INSTRUCTION/DELIVERY Give a demonstration/lecture/simulation/hands-on activity

Show step-by-step solutions to sample problems

Give applications of the theory

Connect to a real-life problem if applicable

4. PRACTICE Provide easy-medium-hard questions

Give time for hands-on unguided classroom work and discovery

Use formative assessment to give feedback

5. ENRICHMENT Provide additional examples and applications

Introduce extensions or generalisations of concepts

Engage in reflection questions

Encourage analysis through higher order thinking prompts

Allow pair/small group discussions

Summarize and synthesize the learnings

6. EVALUATION Supply a diverse question bank for written work and exercises

Provide alternative formats for student work: written homework, journal, portfolio, group/individual projects, student-directed research project

iii

On DepEd Functional Skills and CHEDs College Readiness Standards As Higher Education Institutions (HEIs) welcome the graduates of the Senior High School program, it is of paramount importance to align Functional Skills set by DepEd with the College Readiness Standards stated by CHED.

The DepEd articulated a set of 21st century skills that should be embedded in the SHS curriculum across various subjects and tracks. These skills are desired outcomes that K to 12 graduates should possess in order to proceed to either higher education, employment, entrepreneurship, or middle-level skills development.

On the other hand, the Commission declared the College Readiness Standards that consist of the combination of knowledge, skills, and reflective thinking necessary to participate and succeed - without remediation - in entry-level undergraduate courses in college.

The alignment of both standards, shown below, is also presented in this Teaching Guide - prepares Senior High School graduates to the revised college curriculum which will initially be implemented by AY 2018-2019.

College Readiness Standards Foundational Skills DepEd Functional Skills

Produce all forms of texts (written, oral, visual, digital) based on: 1. Solid grounding on Philippine experience and culture; 2. An understanding of the self, community, and nation; 3. Application of critical and creative thinking and doing processes; 4. Competency in formulating ideas/arguments logically, scientifically,

and creatively; and 5. Clear appreciation of ones responsibility as a citizen of a multicultural

Philippines and a diverse world;

Visual and information literacies Media literacy Critical thinking and problem solving skills Creativity Initiative and self-direction

Systematically apply knowledge, understanding, theory, and skills for the development of the self, local, and global communities using prior learning, inquiry, and experimentation

Global awareness Scientific and economic literacy Curiosity Critical thinking and problem solving skills Risk taking Flexibility and adaptability Initiative and self-direction

Work comfortably with relevant technologies and develop adaptations and innovations for significant use in local and global communities;

Global awareness Media literacy Technological literacy Creativity Flexibility and adaptability Productivity and accountability

Communicate with local and global communities with proficiency, orally, in writing, and through new technologies of communication;

Global awareness Multicultural literacy Collaboration and interpersonal skills Social and cross-cultural skills Leadership and responsibility

Interact meaningfully in a social setting and contribute to the fulfilment of individual and shared goals, respecting the fundamental humanity of all persons and the diversity of groups and communities

Media literacy Multicultural literacy Global awareness Collaboration and interpersonal skills Social and cross-cultural skills Leadership and responsibility Ethical, moral, and spiritual values

iv

The General Mathematics Teaching Guide

Implementing a new curriculum is always subject to a new set of challenges.

References are not always available, and training may be too short to cover all the

required topics. Under these circumstances, providing teachers with quality resource

materials aligned with the curricular competencies may be the best strategy for

delivering the expected learning outcomes. Such is the rationale for creating a series of

teaching guides for several Grade 11 and 12 subjects. The intention is to provide

teachers a complete resource that addresses all expected learning competencies, as

stated in the Department of Educations offi cial curriculum guide.

This resource is a teaching guide for General Mathematics. The structure is quite

unique, re flective of the wide scope of General Mathematics: functions, business

mathematics, and logic. Each lesson begins with an introductory or motivational

activity. The main part of the lesson presents important ideas and provides several

solved examples. Explanations to basic properties, the rationale for mathematical

procedures, and the derivation of important formulas are also provided. The goal is to

enable teachers to move learners away from regurgitating information and towards an

authentic understanding of, and appreciation for, the subject matter.

The chapters on functions are an extension of the functions learned in Junior High

School, where the focus was primarily on linear, quadratic, and polynomial functions.

In Grade 11, learners will be exposed to other types of functions such as piecewise,

rational, exponential, and logarithmic functions. Related topics such as solving

equations and inequalities, as well as identifying the domain, range, intercepts, and

asymptotes are also included.

v

The chapters on business mathematics in Grade 11 may be learners' first opportunity

to be exposed to topics related to financial literacy. Here, they learn about simple and

compound interest, annuities, loans, stocks, and bonds. These lessons can hopefully

prepare learners to analyze business-related problems and make sound financial

decisions.

The final chapter on logic exposes learners to symbolic forms of propositions (or

statements) and arguments. Through the use of symbolic logic, learners should be able

to recognize equivalent propositions, identify fallacies, and judge the validity of

arguments. The culminating lesson is an application of the rules of symbolic logic, as

learners are taught to write their own justifications to mathematical and real-life

statements.

This Teaching Guide is intended to be a practical resource for teachers. It includes

activities, explanations, and assessment tools. While the beginning teacher may use

this Teaching Guide as a script, more experienced teachers can use this resource as a

starting point for writing their own lesson plans. In any case, it is hoped that this

resource, together with the Teaching Guide for other subjects, can support teachers in

achieving the vision of the K to 12 Program.

vi

Hour 1 Hour 2 Hour 3 Hour 4

Week a Lesson 1 Lesson 1, 2 Lesson 3 Lesson 3

Week b Lesson 4 Lesson 5, 6 Lesson 6 Lesson 7

Week c Lesson 7 Lesson 8 Lesson 8 Review/Exam

Week d Lesson 9 Lesson 10 Lesson 10 Lesson 11

Week e Lesson 11 Review/Exam Lesson 12 Lesson 12, 13

Week f Lesson 14 Lesson 14 Lesson 15 Lesson 15

Week g Lesson 16 Lesson 16 Review/Exam Review/Exam

Week h Lesson 17 Lesson 17 Lesson 18, 19 Lesson 19, 20

Week i Lesson 20 Lesson 21 Lesson 21 Lesson 21

Week j Lesson 22 Lesson 22 Review/Exam Review/Exam

First Quarter

Hour 1 Hour 2 Hour 3 Hour 4

Week a Lesson 23 Lesson 24 Lesson 25 Lesson 25, 26

Week b Lesson 26 Lesson 27 Lesson 27 / Review Review/Exam

Week c Lesson 28 Lesson 28 Lesson 29 Lesson 29

Week d Lesson 29 Lesson 30 Lesson 30 Review/Exam

Week e Lesson 31 Lesson 31 Lesson 32 Lesson 33

Week f Lesson 34 Lesson 35 Lesson 35 Review/Exam

Week g Lesson 36 Lesson 36 Lesson 37 Lesson 37

Week h Lesson 38 Lesson 39 Lesson 39 Lesson 39

Week i Lesson 40 Lesson 40 Lesson 40 Lesson 41

Week j Lesson 41 Lesson 41 Review/Exam Review/Exam

Second Quarter

K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL CORE SUBJECT

K to 12 Senior High School Core Curriculum General Mathematics December 2013 Page 1 of 5

Grade: 11 Semester: First Semester Core Subject Title: General Mathematics No. of Hours/Semester: 80 hours/semester Prerequisite (if needed): Core Subject Description: At the end of the course, the students must know how to solve problems involving rational, exponential and logarithmic functions; to solve business-related problems; and to apply logic to real-life situations.

CONTENT CONTENT STANDARDS PERFORMANCE STANDARDS LEARNING COMPETENCIES CODE

Functions and Their Graphs

The learner demonstrates

understanding of...

1. key concepts of functions.

The learner is able to...

1. accurately construct mathematical models to

represent real-life

situations using

functions.

The learner...

1. represents real-life situations using functions, including piece-wise functions.

M11GM-Ia-1

2. evaluates a function. M11GM-Ia-2

3. performs addition, subtraction, multiplication, division, and composition of functions

M11GM-Ia-3

4. solves problems involving functions. M11GM-Ia-4

2. key concepts of rational functions.

2. accurately formulate and solve real-life problems

involving rational

functions.

5. represents real-life situations using rational functions. M11GM-Ib-1 6. distinguishes rational function, rational equation, and

rational inequality. M11GM-Ib-2

7. solves rational equations and inequalities. M11GM-Ib-3 8. represents a rational function through its: (a) table of

values, (b) graph, and (c) equation. M11GM-Ib-4

9. finds the domain and range of a rational function. M11GM-Ib-5 10. determines the:

(a) intercepts

(b) zeroes; and

(c) asymptotes of rational functions

M11GM-Ic-1

11. graphs rational functions. M11GM-Ic-2 12. solves problems involving rational functions,

equations, and inequalities. M11GM-Ic-3




3. key concepts of inverse functions, exponential

functions, and

logarithmic functions.

3. apply the concepts of inverse functions,

exponential functions,

and logarithmic functions

to formulate and solve

real-life problems with

precision and accuracy.

1. represents real-life situations using one-to one functions.

M11GM-Id-1

2. determines the inverse of a one-to-one function. M11GM-Id-2 3. represents an inverse function through its: (a) table of

values, and (b) graph. M11GM-Id-3

4. finds the domain and range of an inverse function. M11GM-Id-4 5. graphs inverse functions. M11GM-Ie-1 6. solves problems involving inverse functions. M11GM-Ie-2 7. represents real-life situations using exponential

functions. M11GM-Ie-3

8. distinguishes between exponential function, exponential equation, and exponential inequality.

M11GM-Ie-4

9. solves exponential equations and inequalities. M11GM-Ie-f-1 10. represents an exponential function through its: (a) table

of values, (b) graph, and (c) equation. M11GM-If-2

11. finds the domain and range of an exponential function. M11GM-If-3 12. determines the intercepts, zeroes, and asymptotes of

an exponential function. M11GM-If-4

13. graphs exponential functions. M11GM-Ig-1 14. solves problems involving exponential functions,

equations, and inequalities. M11GM-Ig-2

15. represents real-life situations using logarithmic functions.

M11GM-Ih-1

16. distinguishes logarithmic function, logarithmic equation, and logarithmic inequality.

M11GM-Ih-2

17. illustrates the laws of logarithms. M11GM-Ih-3 18. solves logarithmic equations and inequalities. M11GM-Ih-i-1 19. represents a logarithmic function through its: (a) table

of values, (b) graph, and (c) equation. M11GM-Ii-2

20. finds the domain and range of a logarithmic function. M11GM-Ii-3 21. determines the intercepts, zeroes, and asymptotes of

logarithmic functions. M11GM-Ii-4

22. graphs logarithmic functions. M11GM-Ij-1 23. solves problems involving logarithmic functions,

equations, and inequalities.

M11GM-Ij-2




Basic Business Mathematics

The learner demonstrates

understanding of...

1. key concepts of simple and compound interests,

and simple and general

annuities.


1. investigate, analyze and solve problems involving

simple and compound

interests and simple and

general annuities using

appropriate business and

financial instruments.

24. illustrates simple and compound interests. M11GM-IIa-1 25. distinguishes between simple and compound interests. M11GM-IIa-2 26. computes interest, maturity value, future value, and

present value in simple interest and compound interest

environment.

M11GM-IIa-b-1

27. solves problems involving simple and compound interests.

M11GM-IIb-2

28. illustrates simple and general annuities. M11GM-IIc-1 29. distinguishes between simple and general annuities. M11GM-IIc-2 30. finds the future value and present value of both simple

annuities and general annuities. M11GM-IIc-d-1

31. calculates the fair market value of a cash flow stream that includes an annuity.

M11GM-IId-2

32. calculates the present value and period of deferral of a deferred annuity.

M11GM-IId-3

2. basic concepts of stocks and bonds.

2. use appropriate financial instruments involving

stocks and bonds in

formulating conclusions

and making decisions.

33. illustrate stocks and bonds. M11GM-IIe-1 34. distinguishes between stocks and bonds. M11GM-IIe-2 35. describes the different markets for stocks and bonds. M11GM-IIe-3 36. analyzes the different market indices for stocks and

bonds. M11GM-IIe-4

37. interprets the theory of efficient markets. M11GM-IIe-5 3. basic concepts of

business and

consumer loans.

3. decide wisely on the appropriateness of

business or consumer

loan and its proper

utilization.

38. illustrates business and consumer loans. M11GM-IIf-1 39. distinguishes between business and consumer loans. M11GM-IIf-2 40. solves problems involving business and consumer loans

(amortization, mortgage). M11GM-IIf-3

Logic The learner demonstrates understanding of...

1. key concepts of propositional logic;

syllogisms and

fallacies.


1. judiciously apply logic in real-life arguments.

41. illustrates a proposition. M11GM-IIg-1 42. symbolizes propositions. M11GM-IIg-2 43. distinguishes between simple and compound

propositions. M11GM-IIg-3

44. performs the different types of operations on propositions.

M11GM-IIg-4

45. determines the truth values of propositions. M11GM-IIh-1 46. illustrates the different forms of conditional

propositions. M11GM-IIh-2

47. illustrates different types of tautologies and fallacies. M11GM-IIi-1




48. determines the validity of categorical syllogisms. M11GM-IIi-2 49. establishes the validity and falsity of real-life arguments

using logical propositions, syllogisms, and fallacies. M11GM-IIi-3

2. key methods of proof and disproof.

2. appropriately apply a method of proof and

disproof in real-life

situations.

50. illustrates the different methods of proof (direct and indirect) and disproof (indirect and by

counterexample).

M11GM-IIj-1

51. justifies mathematical and real-life statements using the different methods of proof and disproof.

M11GM-IIj-2

(x, y) (x, y)

x

y

f = {(1, 2), (2, 2), (3, 5), (4, 5)}

g = {(1, 3), (1, 4), (2, 5), (2, 6), (3, 7)}

h = {(1, 3), (2, 6), (3, 9), . . . , (n, 3n), . . .}

f h x

y g (1, 3) (1, 4)

x y

f g

h

f g x 2 X y 2 Yh X

y x = 7 y = 11 13 x = 2

y = 17 19

x = a (a, b) (a, c)

y x = a

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

x

y x

x y

y = 2x+ 1

y = x2 2x+ 2

x2 + y2 = 1

y =px+ 1

y =2x+ 1

x 1

y = bxc+ 1 bxc

x y x = 0 y +1

1

x

R

R

[1, 1]

[1,+1)

(1, 1) [ (1,+1)

R

y f(x) y

x f

f(x) = 2x+ 1

q(x) = x2 2x+ 2

g(x) =px+ 1

r(x) =2x+ 1

x 1F (x) = bxc+ 1 bxc

C x

40

40 C(x) = 40x

A x

A = xy x

x + 2y = 100 y = (100 x)/2 = 50 0.5xA(x) = x(50 0.5x) = 50x 0.5x2

300

1

m

t(m)

t(m) =

(300 0 < m 100300 +m m > 100

8.00

1.50

d

F (d)

F (d) =

(8 0 < d 48 + 1.5bdc d > 4

bdc dd b4.1c = b4.9c = 4

25

0

0

100

100

100

T (x)

T (x)

15

1, 000

400

f(x) =

8 3

700

f(x) = 700dx4

e x 2 N

150

130 110

100

f(x) =

8>>>>>>>>>>>:

150x 0 x 20

130x 21 x 50

110x 51 x 100

100x x > 100

x 2 N

x

f a

a f f(a)

x = 1.5

f(x) = 2x+ 1

q(x) = x2 2x+ 2

g(x) =px+ 1

r(x) =2x+ 1

x 1F (x) = bxc+ 1 bxc

1.5 x

f(1.5) = 2(1.5) + 1 = 4

q(1.5) = (1.5)2 2(1.5) + 2 = 2.25 3 + 2 = 1.25

g(1.5) =p1.5 + 1 =

p2.5

r(1.5) =2x+ 1

x 1 =2(1.5) + 1

(1.5) 1 =3 + 1

0.5= 8

F (1.5) = bxc+ 1 = b1.5c+ 1 = 1 + 1 = 2

g(4) r(1) g r

4 g(x)r(x)

f q

f(3x 1) q(2x+ 3)

f(3x 1) x f(x) = 2x+ 1 (3x 1)

f(3x 1) = 2(3x 1) + 1 = 6x 2 + 1 = 6x 1

q(3x+ 3) x q(x) = x2 2x+ 2 (2x+ 3)

q(2x+ 3) = (2x+ 3)2 2(2x+ 3) + 2 = (4x2 + 12x+ 9) 4x 6 + 2 = 4x2 + 8x+ 5

f(x) = x 2

f(0) 2

f(3)

f(1)

f() 2

f(x+ 1) x 1

f(3x) 3x 2

f(x) =4

x

f(1) 4

f(2) 2

f(1) 4

f(p2) 2

p2

f(1/x) 4x

f(2x) 2/x

f(x) =px 3

f(3) 0

f(4) 1

f(12) 3

f(x 3)px 6

f

1

1 x

3x 21 x

f(x2 + 4x+ 7)px2 + 4x+ 4 |x+ 2|

200

25

C(x) = 25x+200 x C(x)

2700 3950

t s(t) = 5t2 + 100g = 10m/s2

1

3

2

5

1

3

+

2

5

=

5

15

+

6

15

=

5 + 6

15

=

11

15

1

x 32

x 5

(x 3)(x 5) (x2 8x+ 15)

1

x 3 +2

x 5 =x 5

x2 8x+ 15 +2(x 3)

x2 8x+ 15 =x 5 + 2x 6x2 8x+ 15

=

3x 11x2 8x+ 15

10

21

15

8

10

21

158

=

2 53 7

3 52 2 2 =

6 2 5 6 3 56 3 7 6 2 2 2 =

25

28

x2 4x 5x2 3x+ 2

x2 5x+ 6x2 3x 10

x2 4x 5x2 3x+ 2

x2 5x+ 6x2 3x 10 =

(x+ 1)(x 5)(x 2)(x 1)

(x 2)(x 3)(x 5)(x+ 2)

=

(x+ 1)

(x 5)(x 2)(x 3)

(x 2)(x 1)(x 5)(x+ 2)

=

(x+ 1)(x 3)(x 1)(x+ 2) =

x2 2x 3x2 + x 2

f g

f + g (f + g)(x) = f(x) + g(x)

f g (f g)(x) = f(x)g(x)

f g (f g)(x) = f(x) g(x)

f/g (f/g)(x) = f(x)/g(x)

x g(x) = 0

f(x) = x+ 3

p(x) = 2x 7

v(x) = x2 + 5x+ 4

g(x) = x2 + 2x 8

h(x) = x+ 72 x

t(x) = x 2x+ 3

(v + g)(x)

(f p)(x)

(f + h)(x)

(p f)(x)

(v/g)(x)

(v + g)(x) =x2 + 5x+ 4

+

x2 + 2x 8

= 2x2 + 7x 4

(f p)(x) = (x+ 3) (2x 7) = 2x2 x 21

(f + h)(x) = (x+ 3) +x+ 7

2 x = (x+ 3) 2 x2 x +

x+ 7

2 x =(x+ 3)(2 x) + (x+ 7)

2 x =

=

6 x x2 + x+ 72 x =

13 x2

2 x =13 x2

2 x 11 =

x2 13x 2

(p f)(x) = (2x 7) (x+ 3) = 2x 7 x 3 = x 10

(v/g)(x) = (x2 + 5x+ 4) (x2 + 2x 8) = x2

+ 5x+ 4

x2 + 2x 8

f(x) = 2x+ 1 q(x) = x2 2x+ 2 r(x) = 2x+ 1x 1

f1

(x) = x2 + 3

q(x) f(x)

x2 + 3

q(x) + f(x) = (x2 2x+ 2) + (2x+ 1)

= x2 + 3

= f1

(x)

f1

(x) = q(x) + f(x)

f2

(x) = x24x+1

q(x) f(x) x2 4x+ 1

q(x) f(x) = (x2 2x+ 2) (2x+ 1)

= x2 4x+ 1

= f2

(x)

f2

(x) = q(x) f(x)

f3

(x) =2x2 + x

x 1

2x2 + x

x 1 x 1 r(x) =2x+ 1

x 1f(x) r(x)

f(x) + r(x) = 2x+ 1 +2x+ 1

x 1

=

(2x+ 1)(x 1)x 1 +

2x+ 1

x 1

=

(2x+ 1)(x 1) + (2x+ 1)x 1

=

(2x2 x 1) + (2x+ 1)x 1

=

2x2 + x

x 1= f

3

(x)

(f + g)(x) = f(x) + g(x)

f1

(x) = q(x) + f(x) = (q + f)(x)

f2

(x) = q(x) f(x) = (q f)(x)f3

(x) = f(x) + r(x) = (f + r)(x)

g1

(x) = 2x3 3x2 + 2x + 2

2x3 3x2 + 2x+ 2 f(x) q(x)

f(x) q(x) = (2x+ 1)(x2 2x+ 2)

= (2x)(x2 2x+ 2) + (x2 2x+ 2)

= (2x3 4x2 + 4x) + (x2 2x+ 2)

= 2x3 3x2 + 2x+ 2

= g1

(x)

g1

(x) = f(x) q(x)

g2

(x) = x 1

r(x) =2x+ 1

x 1 x 1 2x + 1f(x) r(x)

f(x)

r(x)= (2x+ 1) 2x+ 1

x 1

= (2x+ 1) x 12x+ 1

=

2x+ 1

2x+ 1 (x 1)

= x 1

= g2

(x)

g2

(x) =f(x)

r(x)

g3

(x) =1

x 1

g3

(x) =1

x 1 r(x) =2x+ 1

x 12x+ 1 r(x) f(x) = 2x+ 1

r(x)

f(x)=

2x+ 1

x 1 (2x+ 1)

=

2x+ 1

x 1 1

2x+ 1

=

1

x 1= g

3

(x)

g3

(x) =r(x)

f(x)

f(x) = 2x+ 1

q(x) = x2 2x+2 (2x+1)2 2(2x+1)+ 2

f g (f g)

(f g)(x) = f(g(x)).

f(x) = 2x+ 1 q(x) = x2 2x+ 2

r(x) = 2x+ 1x 1

g(x) =px+ 1

F (x) = bxc+ 1

(g f)(x)

(g f)(x) = g(f(x))

=

pf(x) + 1

=

p(2x+ 1) + 1

=

p2x+ 2

(q f)(x) (f q)(x)

(q f)(x) = q(f(x))

= [f(x)]2 2 [f(x)] + 2

= (2x+ 1)2 2(2x+ 1) + 2

= (4x2 + 4x+ 1) (4x+ 2) + 2

= 4x2 + 1

(f q)(x) = f(q(x))

= 2(x2 2x+ 2) + 1

= 2x2 4x+ 5

(q f)(x) (f q)(x)

(f r)(x)

(f r)(x) = f(r(x))

= 2r(x) + 1

= 2

2x+ 1

x 1

+ 1

=

4x+ 2

x 1 + 1

=

(4x+ 2) + (x 1)x 1

=

5x+ 1

x 1

(F r)(5)

(F r)(5) = F (r(5))

= br(5)c+ 1

=

2(5) + 1

5 1

+ 1

=

11

4

+ 1 = 2 + 1 = 3

f g f + g f gf g f/g g/f

f(x) = x+ 2 g(x) = x2 4x2 + x 2 x2 + x+ 6 x3 + 2x2 4x 8 1

x 2 x 2

f(x) =px 1 g(x) = x2 + 4px 1 + x2 + 4

px 1 x2 4

px 1(x2 + 4)

px 1

x2 + 4x 2

f(x) =x 2x+ 2

g(x) =1

xx 2x+ 2

+

1

x

x 2x+ 2

1x

x 2x(x+ 2)

x(x 2)x+ 2

x+ 2

x(x 2)

f(x) =1

x+ 2g(x) =

x 2x

1

x+ 2+

x 2x

1

x+ 2 x 2

x

x 2x(x+ 2)

x

(x+ 2)(x 2)(x+ 2)(x 2)

x

f(x) =1

x2g(x) =

px

1

x2+

px

1

x2px

px

x21

x2px

x2px

f(x) = x2 +3x g(x) = x 2f g g f f f g g x2 x 2 x2 + 3x 2 x4 + 6x3 + 12x2 + 9x x 4

(f g) = x(g f) = x

f(x) = 3x 2 g(x) = 13

(x+ 2)

f(x) =x

2 x g(x) =2x

x 1f(x) = (x 1)3 + 2 g(x) = 3

px 2 + 1

p(x) = anxn+ an1x

n1+ an2x

n2+ + a

1

x+ a0

a0

, a1

, . . . , an 2 R an 6= 0 na0

, a1

, a2

, . . . , an

an anxn a0

100, 000

y

x y =100, 000

x

x

y

750 g(x)

g(x) =100, 000

x+ 750

x

y

https://www.desmos.com/calculatorhttps://www.geogebra.org/download

f(x) =p(x)

q(x)p(x) q(x)

q(x) q(x) 6 0f(x) x q(x) 6= 0

v v(t)

t

v t

t

v

v(t) =10

tv t

c(t) =5t

t2 + 1t c(t)

t = 1, 2, 5, 10

t

c(t)

a b c

x y

http://illuminations.nctm.org/Lesson.aspx?id=1968

x

y

y =a

x b + c

x P (x)

P (x) =2x2 + 800

x

b(t) =50t

t+ 10 t 20

t b(t)

t = 1, 2, 5, 10, 15, 20

x2 + 3x+ 2

x+ 4

1

3x2

x2 + 4x 32

px+ 1

x3 1

1

x+ 2x 2

1

(x+ 2)(x 2)

f(x) =p(x)

q(x)p(x) q(x)

q(x)

2

x 3

2x=

1

5

5

x 3 2

xf(x) =

x2 + 2x+ 3

x+ 1

y =x2 + 2x+ 3

x+ 1

x

x y

15px 1

5x4 6x7 + 1 5 x3

x

y = 5x3 2x+ 18

x 8 = x

2x 1px 2 = 4

x 1x+ 1

= x3

y =7x3 4

px+ 1

x2 + 3

6x 5x+ 3

0

x+ 1

2x= 10

x+ 1

2x 10

x2

x 3

2x=

1

5

10x 10x

10x

2

x

10x

3

2x

= 10x

1

5

20 15 = 2x

5 = 2x

x =5

2

xx

x+ 2 1

x 2 =8

x2 4

x

x+ 2 1

x 2 =8

(x 2)(x+ 2)

(x2)(x+2)

(x 2)(x+ 2) xx+ 2

(x 2)(x+ 2) 1x 2 = [(x 2)(x+ 2)]

8

(x 2)(x+ 2)

(x 2)x (x+ 2) = 8

x2 3x 10 = 0

x2 3x 10 = 0

(x+ 2)(x 5) = 0

x+ 2 = 0 x 5 = 0

x = 2 x = 5

x = 2 x = 5

x

x

12 + x 25 + x

12 + x

25 + x= 0.6

25 + x

12 + x

25 + x= 0.6

12 + x = 0.6(25 + x)

12 + x = 0.6(25) + 0.6x

x 0.6x = 15 12

0.4x = 3

x = 7.5

x

60%

v = dt

v = dt t =dv

v v + 10 5v5

v+10

4

3

5

v+

5

v + 10=

4

3

3v(v + 10)

5

v+

5

v + 10=

4

3

3v(v + 10) 5v+ 3v(v + 10) 5

v + 10= 3v(v + 10) 4

3

15(v + 10) + 15v = 4v(v + 10)

30v + 150 = 4v2 + 40v

4v2 + 10v 150 = 0

2v2 + 5v 75 = 0

(2v + 15)(v 5) = 0

v = 152

v = 5

v

(a, b) {x|a < x < b} a b

[a, b] {x|a x b} a b

[a, b) {x|a x < b} a b

(a, b] {x|a < x b} a b

(a,1) {x|a < x}a

[a,1) {x|a x}a

(1, b) {x|x < b}b

(1, b] {x|x b}b

(1,1)R

x

2x

x+ 1 1

2x

x+ 1 1 0

2x (x+ 1)x+ 1

0

x 1x+ 1

0

x = 1 x = 1x = 1

x = 1

1

1 1x 1x+ 1

x < 1 1 < x < 1 x > 1x = 2 x = 0 x = 2

x 1 +x+ 1 + +x 1x+ 1

+ +

x < 1 x 1

1

{x 2 R|x < 1 x 1}(1,1) [ [1,1)

3

x 2 2x = 2 x = 1

2

x = 1 x = 3

2(x+ 1) + + +x + +x 2 +2(x+ 1)

x(x 2) + +

{x 2 R|x < 1 0 < x < 2}

x

h

x h h x

8 = x2h

h x

h =8

x2

h > x

8

x2> x

8

x2> x

8

x2 x > 0

8 x3

x2> 0

(2 x)(x2 + 2x+ 4)x2

> 0

x = 16 x = 28 x = 0 x = 4

4

x < 0 0 < x < 2 x > 2

x = 1 x = 1 x = 32 x + + x2 + 2x+ 4 + + +

x2 + + +(2x)(x2+2x+4)

x2 + +

0 < x < 2

x < 0

x

x

1120

x1600

x+ 4

1600

x+ 4 1120

x 10

1600

x+ 4 1120

x 10

160

x+ 4 112

x 1

160

x+ 4 112

x 1 0

160x 112(x+ 4) (x2 + 4x)x(x+ 4)

0

160x 112x 448 x2 4xx(x+ 4)

0

x2 44x+ 448x(x+ 4)

0

(x 16)(x 28)x(x+ 4)

0

x = 16 x = 28 x = 0 x = 4

x < 4 4 < x < 0 0 < x < 16 16 < x < 28 x > 28x = 5 x = 1 x = 10 x = 20 x = 30

x 16 + +x 28 +x + + +x+ 4 + + + +(x 16)(x 28)

x(x+ 4)+ + +

4 < x < 0 16 < x < 28

x = 16 x = 28

3

x+ 1=

2

x 3

2x

x+ 1+

5

2x= 2 5

x2 10x 1 =

14 5xx 1 4,1

x2 4xx 2 =

14 9xx 2 7

(x+ 3)(x 2)(x+ 2)(x 1) 0 (1,3] [ [2,1)

(x+ 4)(x 3)(x 2)(x2 + 2) 0 [4, 2) [ [3,1)

x+ 1

x+ 3 2 (1,5] [ (3,1)

x 2x2 3x 10 < 0 (1,2) [ (2, 5)

x3 + x

6 + x

f(x) =p(x)

q(x)p(x) q(x)

q(x) q(x) 6 0f(x) x q(x) 6= 0

s = dt

s = 100t

x

s(x)

s(x) =100

x

s = dt

x

x s(x) s(x) =100

x

x

s(x)

s(x) =100

x

https://www.libreoffice.org/download/libreoffice-fresh/http://docs.google.com

f(x) =x 1x+ 1

x x 10 10

x 10 8 6 4 2f(x) 1

1

E F

f(x) =x 1x+ 1

E F

f(x) =x 1x+ 1

x = 1E F

x = 1

E F

x 1

f(x) =x2 3x 10

x

x

x = 0 f

6 x 10 x 6= 0

x 5 4 3 2 1f(x) 6 4.5 2.67 12 6

x 3 4 5 6 7 8 9 10

f(x) 3.33 1.5

x = 0

x 1 x 1

x = 2 x = 5x x

x

p

p(x) =12 + x

25 + x

p(x)

x

p(x)

25 + x

P (t) =

60(t+ 1)

t+ 6

P t bc

t = 5

P (5) =

60(5 + 1)

5 + 6

= b32.726c = 32

P (x)

t

P (t)

t

I

V R I = VR

R

I

f(x) =x 3x+ 4

6 x 2 xx

x 6 5 4 3 2 1f(x) 6 2.5 1.33 0.75 0.4 0.167

f x = 4

f(x) =x2 + x 6x2 + x 20

x = 4,5

6 x 2 x

x 6 5 4 3 2 1f(x) 0.75 0.22 0.3 0.3 0.22 0

x = 3, 2

x

f(x)

x

x

y x = 0

x y

x

f(x) =x 2x+ 2

f(x) {x 2 R | x 6= 2}

x = 2 x = 2f(x) x

x f(x) y 1

x x

x 2 x = 2 x = 2 f(x)x

y f(0) f(0) = 22

= 1

f(x) x

x

x = 2 x 2x < 2 2 x 2 x > 2 2+

x 2

x 3 2.5 2.1 2.01 2.001 2.0001 x 2

f(x) f(x)

f(x) ! +1 x ! 2 f(x)x 2

x 2+

x 1 1.5 1.9 1.99 1.999 1.9999 x 2+

f(x) 3 7 39 399 3999 39999 f(x)

f(x) ! 1 x ! 2+ f(x)x 2

x 2x = 2

x = a f f

x a

a

x = a

x

f(x) x x! +1

f(x) x! +1

x x! +1f(x) f(x) 1

f(x) x x! 1f(x) x! 1

x 5 10 100 1000 10000 x! 1f(x) f(x) 1+

x f(x)

y = 1

y = b f f(x)

b x x! +1 x! 1

y = b b

x 2x! +1 x! 1

x

x = 2x = 2

x < 2 2 < x < 2 x > 2x = 3 x = 0 x = 3

x 2 +x+ 2 + +x 2x+ 2

+ +

x x x

x x = 2

y

f(x) < 1 x! +1 f(x) > 1 x! 1(2, 0)

f(x) x ! 2 f(x)x! 2+

f(x)

y =x 2x+ 2

y = 1 f(x)

f(x) (1, 1) [ (1,+1)

f(x) =4x2 + 4x+ 1

x2 + 3x+ 2

x x! 1 x! +1

x

x = 1000 4x2+4x+1 4, 004, 001

4x2 4, 000, 000

x x2 + 3x + 2 x2

x f(x)4x2

x2= 4 f(x)

x y = 4

f(x) =2x2 5

3x2 + x 7

2x2 53x2 + x 7

2x2

3x2=

2

3

x y = 23

f(x) =3x+ 4

2x2 + 3x+ 1

3x+ 4

2x2 + 3x+ 13x

2x2=

3

2xx x

3

2x0 y = 0

f(x) =4x3 1

3x2 + 2x 5

x4x3 1

3x2 + 2x 54x3

3x2=

4x

3

x4x

3

x

y

n m

n < m y = 0

n = m y = ab ab

n > m

x

y

y

y

x = 0

xx

x

f(x) =3x2 8x 32x2 + 7x 4

x

f(x) (1,4) [ (4, 12

) [ (12

,+1)

f(x)

f(x) =3x2 8x 32x2 + 7x 4 =

(3x+ 1)(x 3)(2x 1)(x+ 4)

y f(0) = 0 0 30 + 0 4 =

3

4

x 3x+ 1 = 0) x = 13

x 3 = 0) x = 3

2x 1 = 0) x = 12

x+ 4 = 0) x = 4

y = 3

2

f(x)

x

4,13

, 12

3

x < 4 4 < x < 13 13 < x 3

x = 10 x = 2 x = 0 x = 1 x = 103x+ 1 + + +x 3 +2x 1 + +x+ 4 + + + +(3x+ 1)(x 3)(2x 1)(x+ 4) + + +

x x x x x

R

x y

x

f(x) =2

x+ 1

f(x) =2

x2 + 2x+ 1

f(x) =3x

x+ 3

f(x) =2x+ 3

4x 7

f(x) =(4x 3)(x 1)(2x+ 1)(x+ 1)

f(x) =(5x 2)(x 2)(3x 4)(x+ 2)

f(x) =x2 x+ 6x2 6x+ 8

f(x) =x2 4x 5

x 4

f(x) =x 1

x3 4x

f(x) =x2 9x2 + 4

N(t) t

N(t) =75t

t+ 5t 0.

N

N(t) t!1t ! 1 N(t) ! 75

c

t

c(t) =20t

t2 + 2t 0.

c(t) c(t) t!1t!1 c(t)! 0

x = 3 x = 3 y = 1 x 5 y5

9

f(x) =(x 5)2(x2 + 1)

(x+ 3)(x 3)(x2 + 5)

p =5125000V 2 449000V + 19307

125V 2(1000V 43)p V

p V

V = 0 V = 43/1000

p = 0

p

V

V = 0.043

TC =5

9

(TF 32)

TF =9

5

TC + 32

F

C

200

F 93.33C 120C 248F

f x1

x2

f f(x1

) 6= f(x2

)

y x

2

d

F (d) =

(8.00 0 < d 4(8.00 + 1.50 bdc) d > 4

bdc d

F (3) = 8

F (3) = F (2) = F (3.5) = 8 F

y x

https://www.world-airport-codes.com

y x 2

y = x24

y = 2x 1

x 4 3 2 1y 9 7 5 3 1

y x

x y

x 9 7 5 3 1y 4 3 2 1

x

y

x 4 3 2 1y 1 1 1 1 0

y

x y = 1 x = 1, 2, 3, 4

x y

x 1 1 1 1 0y 4 3 2 1

x = 1 y

x y

x y

f A B f

f1 B A f1(y) = x

f(x) = y y B

x y

y

x

y = f(x)

x y

y x

f(x) = 3x+ 1

y = 3x+ 1

x y x = 3y + 1

y x

x = 3y + 1

x 1 = 3yx 13

= y =) y = x 13

f(x) = 3x+ 1 f1(x) =x 13

f(f1(x)) f1(f(x))

f(x) f1(x)

f1(x) f(x)

f(f1(x)) = x x f1

f1(f(x)) = x x f

g(x) = x3 2

y = x3 2x y x = y3 2

y x

x = y3 2

x+ 2 = y3

3px+ 2 = y =) y = 3

px+ 2

g(x) = x3 2 g1(x) = 3px+ 2

f(x) =2x+ 1

3x 4

y =2x+ 1

3x 4x y x =

2y + 1

3y 4y x

x =2y + 1

3y 4x(3y 4) = 2y + 1

3xy 4x = 2y + 1

3xy 2y = 4x+ 1 y

y

y(3x 2) = 4x+ 1

y =4x+ 1

3x 2

f(x) f1(x) =4x+ 1

3x 2

f(x) = x2 + 4x 2

y = x2 + 4x 2

x y x = y2 + 4y 2y x

x = y2 + 4y 2

x+ 2 = y2 + 4y

x+ 2 + 4 = y2 + 4y + 4

x+ 6 = (y + 2)2

px+ 6 = y + 2

px+ 6 2 = y =) y =

px+ 6 2

y = px+ 6 2 x

y x = 3 y 1 5f(x) = x2 + 4x 2

f(x) = |3x|

y = |3x|

f1 f

f(1) = f(1) = 3 x 1 y f

y = |4x|x y x = |4y|

y x

x = |4y|

x =p(4y)2 |x| =

px2

x2 = 4y2

x2

4

= y2

r

x2

4

= y =) y = r

x2

4

x = 2 y = 1 y = 1 y = q

x24

f(x) = |3x|

k(t) = 59

(t32)+273.15t

k = 59

(t 32) + 273.15k t

t k

k =5

9

(t 32) + 273.15

k 273.15 = 59

(t 32)9

5

(k 273.15) = t 329

5

(k 273.15) + 32 = t =) t = 95

(k 273.15)

t(k) = 95

(k 273.15) k

f(x) =1

2

x+ 4 f1(x) = 2x 8

f(x) = (x+ 3)3 f1(x) = 3px 3

f(x) =3

x 4 f1

(x) =4x+ 3

x

f(x) =x+ 3

x 3 f1

(x) =3x+ 3

x 1

f(x) =2x+ 1

4x 1 f1

(x) =x+ 1

4x 2

f(x) = |x 1|

y = x

y = x

y = x

y = x

http://artforkidshub.com/5-free-symmetry-art-activity/

y = x

y = x

x y

f f1 f1(f(x)) = x f1(x)

y f(x) x

xf(x)! y f

1(y)! x

y = x

y = f1(x) y = f(x) = 2x + 1

{x |2 x 1.5} f(x)

y = 2x+ 1 y = x

{y 2 R | 3 y 4}

f1(x) =x 12

f1(x) = [3, 4]

f1(x) = [2, 1.5]

f(x) f1(x)

[2, 1.5] [3, 4][3, 4] [2, 1.5]

f(x) =1

x

f(x) =1

xy = x

y = x f(x) f1(x) = f(x)

f1(x) = f(x) =1

x

f(x) = 3px+ 1

f(x) y = x

f(x) = 3px+ 1 y = x3 1

f1(x) = x3 1

f(x) =5x 1x+ 2

f(x) = (1, 2) [ (2,1)

f(x) = (1,5) [ (5,1)

x = 2

y = 5

y = x

x y

y = x

x = 5y = 2

f(x) f1(x)

(1, 2) [ (2,1) (1,5) [ (5,1)(1,5) [ (5,1) (1, 2) [ (2,1)

y = x

y = x

f1

1(x) = f(x)

f(x) = (x+ 2)2 3 2 = 3(x+ 2)2

2

x 0

x 0

x y x =3(y + 2)2

2

, y 0y x

x =3(y + 2)2

2

2x

3

= (y + 2)2

r2x

3

= y + 2 y 2 r

2x

3r2x

3

2 = y =) y =r

2x

3

2

x = 54

f1(54) =

r2(54)

3

2 =r

108

3

2 =p36 2 = 6 2 = 4

t

d

t(d) =

12.5

d

3

t =

12.5

d

3

d t

d t

t =

12.5

d

3

3pt =

12.5

d

d =12.53pt

d(t) =12.53pt

t = 6.5 d(6.5) =22.53p6.5

= 12.06

12.06

f(x) = x2 + 1

{0, 0.5, 1, 1.5, 2, 2.5, 3}

x 0 0.5 1 1.5 2 2.5 3

f(x) 1 1.25 2 3.25 5 7.25 10f

x 1 1.25 2 3.25 5 7.25 10

f1(x) 0 0.5 1 1.5 2 2.5 3

A = {(4, 4), (3, 2), (2, 1), (0,1), (1,3), (2,5)}

A1 = {(4,4), (2,3), (1,2), (1, 0), (3, 1), (5, 2)}

f(x) = 2px 2 + 3

[2,1)

[3,1) [2,1)

f(x) =3x+ 2

x 4(1, 3) [ (3,1)

(1, 4) [ (4,1)

x = a f1(a) = 0 1/2

w l

w = (3.24 103)l2

l 0

l =p

w/(3.4 103)

n

s n = 2s

b f(x) = bx y = bx

b > 0 b 6= 1

x = 3, 2, 1, 0, 1, 2 3

y =1

3

xy = 10x y = (0.8)x

x 3 2 1

y =1

3

x 13

1

9

1

27

y = 10x1

1000

1

100

1

10

y = (0.8)x

f(x) = 3x f(2) f(2) f1

2

f(0.4) f()

f(2) = 32 = 9

f(2) = 32 = 13

2

=

1

9

f

1

2

= 3

1/2=

p3

f(0.4) = 30.4 = 32/5 =5p3

2

=

5p9

3.14159 3

f() = 3 33.14 33.14159

3

3

b

b

g(x) = a bxc + d

a c d

t t = 0

t

t = 0

t = 100 20(2)

t = 200 = 20(2)2

t = 300 = 20(2)3

t = 400 = 20(2)4

y = 20(2)t/100

y T y0

y t y = y0

(2)

t/T

t

t = 0

t = 10

t = 20

t = 30

y = 101

2

t/10

100, 000 6%

t

t = 0 = 100, 000

t = 1 = 100, 000(1.06) = 106, 000

t = 2 = 106, 000(1.06) = 112, 360

t = 3 = 112, 360(1.06) 119, 101.60t = 4 = 119, 101.60(1.06) 126, 247.70t = 5 = 26, 247.70(1.06) 133, 822.56

y = 100, 000(1.06)t

P r

t A = P (1 + r)t

y = 100000(1.06)t t t

t = 8 y = 100, 000(1.06)8 159, 384.81 t = 10 y = 100, 000(1.06)10 179, 084.77 200, 000

e 2.71828 ee

e

f(x) = ex

T t

T = 170165e0.006t t

t

T

50, 000

A = 50, 000(1.044)t t

t = 18 A = 50, 000(1.044)18 108, 537.29100, 000

A = 20, 000(1.05)t t

t = 10, A = 20, 000(1.05)10 32, 577.8932, 577.89

y = 1001

2

x/250x = 0

x = 500 y = 1001

2

500/250

= 100

1

2

2

= 25

y = 1, 000(3)x/80 x = 0

x = 100 y = 1, 000(3)100/80 = 3, 948.22 3, 948

10, 000

A = 10, 000(1.02)t t

A = 10, 000(1.02)12 = 12, 682.42

12, 682.42

4

x1= 16x y = 2x 2x 26

a bxc + d b > 0b 6= 1

f(x) = bx b > 0

b 6= 1

7

2xx2=

1

343

5

2x 5x+1 0f(x) = (1.8)x y =

(1.8)x

x

x y

f(x) = 2x3

f(x) = 2x

y = ex

2

2

(5

x+1) = 500

625 5x+8

a 6= 0a0 = 1

an =1

an

r s

aras = ar+s

ar

as= ars

(ar)s = ars

(ab)r = arbrab

r=

ar

br

49 = 7

x+1

7 = 2x+ 3

3

x= 3

2x1

5

x1= 125

8x = x2 9x2 = 3x3 + 2x 12x+ 3 > x 12

x2 > 8

x1

6= x2

bx1 6= bx2 bx1 = bx2 x1

= x2

4

x1= 16

4

x1= 16

4

x1= 4

2

x 1 = 2

x = 2 + 1

x = 3

4

x1= 16

(2

2

)

x1= 2

4

2

2(x1)= 2

4

2(x 1) = 4

2x 2 = 4

2x = 6

x = 3

x = 3 431 = 42 =

16

125

x1= 25

x+3

125

x1= 25

x+3

(5

3

)

x1= (5

2

)

x+3

5

3(x1)= 5

2(x+3)

3(x 1) = 2(x+ 3)

3x 3 = 2x+ 6

x = 9

9

x2= 3

x+3

(3

2

)

x2= 3

x+3

3

2x2= 3

x+3

2x2 = x+ 3

2x2 x 3 = 0

(2x 3)(x+ 1) = 0

2x 3 = 0 x+ 1 = 0

x =3

2

x = 1

b > 1 y = bx x

bx < by x < y

0 by x < y

bm < bn

m < n m > n b

3

x < 9x2

3

x < (32)x2

3

x < 32(x2)

3

x < 32x4

3 > 1

x < 2x 4

4 < 2x x

4 < x

(4,+1] x = 5 x = 4

1

10

x+5

1

100

3x

1

100

=

1

10

2

1

10

1

10

x+5

1

100

3x

1

10

x+5

1

10

2

3x

1

10

x+5

1

10

6x

1

10

< 1

x+ 5 6x

5 6x x

5 5x

1 x

[1,+1) x = 1 x = 0 1

y0

1

256

t

y = y0

1

2

t/2.45y0

1

2

t/2.45=

1

256

y0

1

2

t/2.45=

1

256

1

2

t/2.45=

1

2

8

t

2.45= 8

t = 19.6

t = 0

(0.6)x3 > (0.36)x1

(0.6)x3 > (0.36)x1

(0.6)x3 > (0.62)x1

(0.6)x3 > (0.6)2(x1)

(0.6)x3 > (0.6)2x2

x 3 > 2x 2

3x > 1

x >1

3

(0.6)x3 > (0.36)x1

(0.6)x3 > (0.62)x1

(0.6)x3 > (0.6)2(x1)

(0.6)x3 > (0.6)2x2

x 3 < 2x 2

3x < 1

x (0.6)2x2 x 3 > 2x 2x > 1

3

x

16

2x3= 4

x+2 x = 83

1

2

2x

= 2

3x 1

2

2

1 x = 3

4

2x+7 322x329

6

,+1

2

5

5x1

254

25

4

2

5

2(1,1

5

]

x

7

x+4= 49

2x1 x = 2

4

x+2= 8

2x x = 12

3

5x+2

=

3

2

2x

27

,+1

f(x) = bx b > 1

f(x) = bx 0 < b < 1

f(x) = 2x

f(x)

x 4 3 2 1

f(x)1

16

1

8

1

4

1

2

f(x) = 2x f(x) = 2x

x

y x

y = 0

g(x) =

1

2

x

g(x)

x 3 2 1

f(x)1

2

1

4

1

8

1

16

g(x) =

1

2

x

g(x) =

1

2

x

x

x y = 0

b > 1 0 1 0 < b < 1

f(x) = bx b > 0 b 6= 1

R

(0,+1)

y x

y = 0 x

f(x) = 2x g(x) = 3x

x 4 3 2 1f(x)

g(x)

x 4 y 1f(x) g(x)

y

f(x) =

1

2

xg(x) =

1

3

x

x 4 3 2 1f(x)

g(x)

x 4 y 1f(x) g(x)

y

f(x) = 5x y

5

x=

1

5

x=

1

5

x

x 3

x 4 y1

y = 2x

y = 2x

y = 2xy = 3 2x

y = 25

2xy = 2x + 1

y = 2x 1y = 2x+1

y = 2x1

x 3 2 1y = 2x

y = 2x

y = 2x 0.125 .25 .5 1 2 4 8y

1

y = 3 2xy

y = 25

2xy

2/5

y = 2x + 1

y = 2x 1 0.875 0.75 0.5

y = 2x+1

y = 2x1

y = 2x y = 2x y = 2x

y

x 3 2 1y = 2x

y = 2x 0.125 0.25 0.5 1 2 4 8y = 2x

y y = 2x yy = 2x y = 2x y = 2x

x

y = 2x x y = 2x x y = 2x

y = 2x y

y = f(x) x y = f(x)

y = f(x) y y = f(x)

y = 2x y = 3(2x) y = 0.4(2x)

y

x 3 2 1y = 2x

y = 3(2x)

y = 0.4(2x)

y y = 3(2x) y

y = 2x y y = 0.4(2x)

y y = 2x

R

y y y = 3(2x)

y y = 0.4(2x)

y = 0

(0,+1)

a > 0 y = af(x) y

y = f(x) a

a > 1 0 < a < 1 y = f(x)

y = 2x y = 2x 3 y = 2x + 1

y

x 3 2 1y = 2x

y = 2x 3 2.875 2.75 2.5 2 1y = 2x + 1

R

y = 2x + 1 (1,+1) y = 2x 3 (3,+1)

y y

y = 2x

y = 2x y = 0

y = 2x + 1 y = 1 y = 2x 3y = 3

d y = f(x) + d d

d > 0 d d < 0 y = f(x)

y = 2x y = 2x2 y = 2x+4

y

x 3 2 1y = 2x

y = 2x2

y = 2x+4

R

(0,+1)

y x = 0 y

y = 2x+4 24 = 16 y y = 2x2 22 = 0.25

y = 0

c y = f(x c) cc > 0 c c < 0

y = f(x)

f(x) = a bxc + d

b b > 1 0 0 d d < 0

c c > 0 c c < 0

y = bx

y

y = 3x 4

y =

1

2

x+ 2

y = 2x5

y = (0.8)x+1

y = 2

1

3

x

y = 0.25(3x)

y = 2x3 + 1

y =

1

3

x1 2

x

2

4

= x 43 = x 51 = x16

12= x

1

5

1

4

5

x= 625

3

x=

1

9

7

x= 0

10

x= 100, 000

x

x b logb x b

x log3

81 = 4 3

4

= 81

log

2

32 = 5 2

5

= 32

log

5

1 = 0 5

0

= 1

log

6

1

6

= (1) 61 = 1

6

log

2

32

log

9

729

log

5

5

log 1216

log

7

1

log

5

1p5

4

1p5

1/2

a b b 6= 1 ab logb a b

logb a= a logb a

b a

log

2

32 = 5 2

5

= 32

log

9

729 = 3 9

3

= 729

log

5

5 = 1 5

1

= 5

log

1/2 16 = 41

2

4= 16

log

7

1 = 0 7

0

= 1

log

5

1p5

= 12

5

1/2=

1p5

logb a = c

bc = a

b c c = logb a

logb a a log2(8)

logb x log51

125

= 3 53 = 1125

log x log10

x

e

e

ln lnx loge x

5

3

= 125

7

2=

1

49

10

2

= 100

2

3

2

=

4

9

(0.1)4 = 10, 000

4

0

= 1

7

b= 21

e2 = x

(2)2 = 4

log

5

125 = 3

log

7

1

49

= 2

log 100 = 2

log 23

4

9

= 2

log

0.1 10, 000 = 4log

4

1 = 0

log

7

21 = b

b = 3 73 6= 21 b7

1.5645

lnx = 2

logm = n

log

3

81 = 4

log

p5

5 = 2

log 34

64

27

= 3

log

4

2 =

1

2

log

10

0.001 = 3ln 8 = a

10

n= m

3

4

= 81

(

p5)

2

= 53

4

3=

64

27

4

1/2= 2

10

3= 0.001 103 =

1

1, 000ea = 8

10

31

log 10

31

= 31

R =2

3

log

E

10

4.40

E 104.40

10

12

E = 1012 R =2

3

log

10

12

10

4.40=

2

3

log 10

7.6

log107.6 = 7.6

10

7.6log 10

7.6= 7.6

R =2

3

(7.6) 5.1

10

12/104.40 = 107.6 39810717

http://www.phivolcs.dost.gov.ph/index.php?option=com_content&task=view&id=45&Itemid=100

D = 10 logI

10

12

I 2 1012 2

10

6

60 85

90 100

10

6 m2

D = log10

6

10

12 = 10 log 106

log 10

6

10

6

log 10

6

= 6

D = 10(6) = 60

10

6

10

12 = 106

= 100, 000

[H+]

pH = log[H+]

pH = log1

[H+]

10

5 log 105 log 105

10

5log 10

5= 5

log 105 = (5) = 5

log

3

243 5

log

6

1

216

3

log

0.25 16 2

49

x= 7 log

49

7 = x

6

3=

1

216

log

6

216 = 3

10

2

= 100 log 100 = 2

log 112

4

121

= 2

11

2

2=

4

121

ln 3 = y ey = 3

log 0.001 = 3 103 = 0.001

log

3

(x 2) = 5 lnx 9 y = log 12x

logx 2 = 4 lnx2 > (lnx)2 g(x) = log

3

x

x

x y

logx 2 = 4

log2

x = 4

log2

4 = x

logx 2 = 4 x

log2

x = 4 4

log2

4 = x x

log2 x = 4

f(x) = 3x

x f(x)

41

g(x) = log3

x

x

x g(x)1

81

1

3

log

2

x = 4 logx 16 = 2 log 1000 = x

log

2

x = 4

2

4

= x

x = 16

logx 16 = 2

x2 = 16

x2 16 = 0(x+ 4)(x 4) = 0x = 4,+4

x = 4

log 1000 = x

log 1000

3 = xx = 3

log

3

81

3

4

= 81

log

4

1 log

3

3 log

4

4

2

log

4

3 = 0 4

0

= 1

log

3

3 = 1 3

1

= 3

log

4

4

2

= 2 4

2

= 4

2

b x b > 0 b 6= 1

logb 1 = 0

logb bx= x

x > 0 blogb x = x

logb 1 b b?

= 1

logb bx b bx

x

logb x b x b

x

log

2

14

2

3

= 8 2

4

= 16

2

3.8074 14.000

2

log2 14= 14

logb 1 = 0 logb bx= x blogb x = x

log 10

ln e3

log

4

64

log

5

1

125

5

log5 2

log 1

log 10 = log

10

10

1

= 1

ln e3 = loge e3

= 3

log

4

64 = log

4

4

3

= 3

log

5

1

125

= log

5

5

3= 3

5

log5 2=2

log 1 = 0

10

2 m2

D = 10 log

I

I0

D = 10 log

10

2

10

12

D = 10 log 1010

D = 10 10D = 100

pH = log[H+]3.0 = log[H+]3.0 = log[H+]10

3.0= 10

logH+

10

3.0= [H+]

10

3.0

log

7

49

log

27

3

1

3

ln e

log

7

7

3 78

log

7

7

3

+ log

7

7

8

log

7

49

7

log

7

49 log7

7

log

7

7

5

5 log7

7

log

2

2

4

2

10

log

2

2

4 log2

2

10 6

log

3

(27 81) log3

27 + log

3

81

b > 0, b 6= 1 n 2 R u > 0, v > 0

logb(uv) = logb u+ logb v

logb

uv

= logb u logb v

logb(un) = n logb u

logb(uv) = logb u+ logb v log77

3 78= log

7

7

3

+ log

7

7

8

logb

uv

= logb u logb v

log

7

49

7

= log

7

49 log7

7

log

2

2

4

2

10

= log

2

2

4 log2

2

10

logb(un) = n logb u

log

7

7

5

= 5 log7

7

log

3

(27 81) = log3

27 + log

3

81

r = logb u s = logb v u = br v = bs

logb(uv) = logb(brbs)

) logb(uv) = logb br+s

) logb(uv) = r + s

) logb(uv) = logb u = logb v

r = logb u u = br un = brn

un = brn

) logb(un) = logb(brn)

) logb(un) = rn

) logb(un) = n logb u

logb un

(logb u)n n

u n logb u

log

2

(5 + 2) 6= log2

5 + log

2

2

log

2

(5 + 2) 6= (log2

5)(log

2

2)

log

2

(5 2) 6= log2

5 log2

2

log

2

(5 2) 6= log2 5log

2

2

log

2

(5

2 2) 6= 2 log2

(5 2)

log(ab2)

log

3

3

x

3

ln[x(x 5)]

log(ab2) = log a+ log b2 = log a+ 2 log b

log

3

3

x

3

= 3 log

3

3

x

= 3(log

3

3 log3

x) = 3(1 log3

x) = 3 3 log3

x

ln[x(x 5)] = lnx+ ln(x 5)

log 2 + log 3

2 lnx ln ylog

5

(x2) 3 log5

x

2 log 5

log 2 + log 3 = log(2 3) = log 6

2 lnx ln y = ln(x2) ln y = lnx2

y

log

5

(x2) 3 log5

x = log5

(x2) log5

(x3) = log5

x2

x3

= log

5

1

x

= log

5

(x1) = log5

x

2 log 5

2 = 2(1)

= 2(log 10) 1 = logb b

= log 10

2 n logb u = logb un

= log 100

2 log 5 = log 100 = log100

5

= log 20

log

3

729 6

log

9

729 3

log

27

729 2

log

1/27 729 2log

729

729 1

log

81

729 3/2

log

81

729

81

1/4= 3 3

6

= 729 (81

1/4)

6

= 729

81

6/4= 81

3/2= 729 log

81

729 =

3

2

log

3

n 729 =6

n

a b x a 6= 1, b 6= 1logb x =

loga x

loga b

log

3

n 729 =6

n

log

3

n 729 =log

3

729

log

3

3

n=

6

n.

log

8

32

log

243

1

27

log

25

1p5

log

8

32 =

log

2

32

log

2

8

=

5

3

log

243

1

27

=

log

3

1

27

log

3

243

=

35

log

25

1p5

=

log

5

1p5

log

5

25

=

1/22

= 14

log

6

4

log 122 e

log

6

4 =

log

2

4

log

2

6

=

2

log

2

6

log 122 =

ln 2

ln

1

2

=

ln 2

ln 1 ln 2 =ln 2

0 ln 2 =ln 2

ln 2 = 1

log

x3

2

3 log x log 2

ln(2e)2 (2e)2 = 4e3 2 ln 2 + 2

log

4

(16a) 2 + log4

a

log(x+ 2) + log(x 2) log(x2 4)

2 log

3

5 + 1 log

3

75

2 ln

3

2

ln 4 ln

9

16

(log

3

2)(log

3

4) = log

3

8

(log

3

2)(log

3

4) = log

3

6

log 2

2

= (log 2)

2

log

4

(x 4) = log4 xlog

4

4

3 log

9

x2 = 6 log9

x

3(log

9

x)2 = 6 log9

x

log

3

2x2 = log3

2 + 2 log

3

x

log

3

2x2 = 2 log3

2x

log

5

2 0.431

log

5

8, log5

1

16

, log

5

p2, log

25

2 log

25

8

1.2920,1.7227, 0.2153, 0.2153, 0.6460

log 6 0.778 log 4 0.602

log

6

4, log 24, log4

6, log

2

3

log

3

2

0.7737, 1.3802, 1.2925,0.1761, 0.1761

b x logb x

b

x

logb x

f(x) = logb x logb u = logb v u = v

ab = 0 a = 0 b = 0

x

log

4

(2x) = log4

10

log

3

(2x 1) = 2logx 16 = 2

log

2

(x+ 1) + log2

(x 1) = 3log x2 = 2

(log x)2 + 2 log x 3 = 0

log

4

(2x) = log4

10

2x = 10

x = 5

log

4

(2 5) = log4

(10)

log

3

(2x 1) = 22x 1 = 32

2x 1 = 92x = 10

x = 5

log

3

(2 (5)1) = log3

(9) = 2

logx 16 = 2

x2 = 16

x2 16 = 0(x+ 4)(x 4) = 0 a2 b2 = (a+ b)(a b)x = 4, 4

log

4

(16) = 2 4 log4(16)

log

2

(x+ 1) + log2

(x 1) = 3log

2

[(x+ 1)(x 1)] = 3 logb u+ logb v = logb(uv)(x+ 1)(x 1) = 23

x2 1 = 8x2 9 = 0(x+ 3)(x 3) = 0 a2 b2 = (a+ b)(a b)x = 3, 3

log

2

(3 + 1) log

2

(31

3 log2

(3 + 1) = log

2

(2)

x = 3

log x2 = 2

log x2 = 2

x2 = 102

x2 = 100

x2 100 = 0(x+ 10)(x 10) = 0x = 10, 10

log(10)

2

= 2 log(10)

2

= 2

log x2 = 2

log x2 = log 102 2 = 2(1) = 2(log 10) = log 102

x2 = 102

x2 100 = 0(x+ 10)(x 10) = 0x = 10, 10

log(10)

2

= 2 log(10)

2

= 2

logb un= n logb u

log x2 = 2

2 log x = 2

log x = 1

x = 10

log x2 = 2 log x x > 0

(log x)2 + 2 log x 3 = 0

log x = A

A2 + 2A 3 = 0(A+ 3)(A 1) = 0A = 3 A = 1

log x = 3 log x = 1x = 103 =

1

1000

x = 10

log

1

1000

log 10

x 2x = 3

2

x= 3

log 2

x= log 3

x log 2 = log 3 logb un= n logb u

x =log 3

log 2

1.58496

x log 12x

1

8

1

4

1

2

x log2 x

1

8

1

4

1

2

x log 12x

1

8

1

4

1

2

123

x log2

x

1

8

31

4

21

2

1

1

2

log 12x x log 1

2x

log

2

x x log2

x

logb x

0 logb x2

b > 1 x1

< x2

logb x1 < logb x2

< > b

x

log

3

(2x 1) > log3

(x+ 2)

log

0.2 > 32 < log x < 2

log

3

(2x 1) > log3

(x+ 2)

2x 1 > 0 x+ 2 > 02x 1 > 0 x > 1

2

x+ 2 > 0 x > 2x > 1

2

x > 12

x 2

log

3

(2x 1) > log3

(x+ 2)

2x 1 > x+ 2x > 3 x

) x > 3(3,+1)

log

0.2 > 3

x > 0

3 15

3 = log 15

1

5

3

log 15x > log 1

5

1

5

3

0.2 =1

5

log 15x > log 1

5

1

5

3x 0

2 log 102 log 102

log 10

2 < log x < log 102

log 10

2 < log x log x < log 102

10

2 < x x < 102

1

100

< x < 100

1

100

, 100

EB ENEBEN

7.2 =2

3

log

EB10

4.46.7 =

2

3

log

EN10

4.4

EB 7.2

3

2

= log

EB10

4.4

10.8 = logEB10

4.4

10

10.8=

EB10

4.4

EB = 1010.8 104.4 = 1015.2

EN 6.7

3

2

= log

EN10

4.4

10.05 = logEN10

4.4

10

10.05=

EN10

4.4

EB = 1010.05 104.4 = 1014.45EBEN

=

10

15.2

10

14.45= 10

0.75 5.62

n

n+ 1

E1

E2

n n+1E

2

E1

n =2

3

log

E1

10

4.4n+ 1 =

2

3

log

E2

10

4.4

E1

3

2

n = logE

1

10

4.4

10

3n/2=

E1

10

4.4

E1

= 10

3n/2 104.4 = 103n2 +4.4

E2

3

2

(n+ 1) = logE

2

10

4.4

10

3(n+1)/2=

E2

10

4.4

E2

= 10

3(n+1)/2 104.4 = 103(n+1)

2 +4.4

E2

E1

=

10

3(n+1)2 +4.4

10

3n2 +4.4

= 10

32 31.6

A = P (1 + r)n A

P r n

A = 2P r = 2.5% = 0.025

A = P (1 + r)n

2P = P (1 + 0.025)n

2 = (1.025)n

log 2 = log(1.025)n

log 2 = n log(1.025)

n =log 2

log 1.025 28.07

log ln

ln 2

ln 1.025

P (x) = 20, 000, 000 e0.0251x x x = 0

P (x) = 200, 000, 000

200, 000, 000 = 20, 000, 000 e0.0251x

10 = e0.0251x

ln 10 = ln e0.0251x

ln 10 = 0.0251x(ln e)

ln 10 = 0.0251x

x =ln 10

0.0251 91

f(t) = Aekt

t A k

f(0) = 5, 000 f(90) = 12, 000

f(0) = Aek(0) = A = 5, 000

f(90) = 5, 000ek(90) = 12, 000) e90k = 125

ln e90k = ln12

5

) 90k = ln 125

) k 0.00973f(t) = 5, 000 e0.00973t

f(180) = 5, 000 e0.00973(180) 28, 813

y =ex + ex

2

y = 4

http://newsinfo.inquirer.net/623749/philippines-welcomes-100-millionth-babyhttp://mathworld.wolfram.com/Catenary.htmlhttp://mathworld.wolfram.com/Catenary.html

4 =

ex + ex

2

8 = ex + ex

8 = ex + ex

8 = ex +1

ex8ex = e2x + 1

e2x 8ex + 1 = 0u = ex u2 = e2x u2 8u+ 1 = 0

u = 4p15

u = ex 4 +p15 = ex 4

p15 = ex

ln(4 +

p15) = x ln(4

p15) = x

y = 4

x ln(4 +p15) ln(4

p15) 4.13

log

5

(x 1) + log5

(x+ 3) 1 = 0

log

3

x+ log3

(x+ 2) = 1

3

x+1= 10

1 log 3log 3

1.0959

lnx > 1 e 2.7183 (e,+1)

log

0.5(4x+ 1) < log0.5(1 4x) (0, 1/4)

log

2

[log

3

(log

4

x)] = 0

f(x) = bx

f(x) = bx

y = f(x) = bx

y = bx

x = by x y

y = logb x

f1(x) = logb x

y = log2

x

y = log2

x

x 116

1

8

1

4

1

2

y 4 3 2 1

y = log2 x y = log2 x

x > 0

x

x = 0

y = 2x y = log2

x

y = x

y = log 12x

y = log 12x

x 116

1

8

1

4

1

2

y 4 3 2 1 1 2 3

y = log 12x y = log 1

2x

x > 0

x

x = 0

y = log2

x y = log 12x

y = logb x b

y = logb

x(b > 1) y = logb

x(0 1 0 < b < 1

{x 2 R|x > 0}x logb x

x y

x = 0 y

y = x

y = bx y = logb

x(b > 1) y = bx y = logb

x(0 < b < 1)

y = log2

x

y = log 12x

y = 2 log2

x

x

y = 2 log2

x y = 2 log2

x

y

x 116

1

8

1

4

1

2

log

2

x 4 3 2 1y = 2 log

2

x 8 6 4 2

{x|x 2 R, x > 0}

{y|y 2 R}

x = 0

x

y = log3

x 1

y = log3

x >

1y = log

3

x (1, 0), (3, 1) (9, 2)

(1,1), (3, 0) (9, 1)

{x|x 2 R, x > 0}

{y|y 2 R}

x = 0

x

x x

y = 0

0 = log

3

x 1log

3

x = 1

x = 31 = 3

y = log0.25(x+ 2)

y = log0.25 x 0 < 0.25 < 1

y = log0.25[x (2)]

2y = log

0.25 x (1, 0), (4,1), (0.25, 1)(1, 0), (2,1), (1.75, 1)

{x|x 2 R, x > 2}x + 2 log

0.25(x + 2) x

2{y|y 2 R}

x = 2x 1

f(x) = a logb(x c) + d

b b > 1 0 0 dd < 0 c c > 0 c

c < 0

y = logb x y

y = logx(x+ 3)

y = log 13(x 1)

y = (log5

x) + 6

y = (log0.1 x) 2

y = log 25(x 4) + 2

y = log6

(x+ 1) + 5

log 2 0.3010 log 3 0.4771 log 5 0.6990 log 7 0.8451

2

1/3

5

1/4

2

1/3

5

1/4

n =2

1/3

5

1/4log n =

1

3

log 2 14

log 5

log n

log n 13

(0.3010) 14

(0.6990) 0.0744

0.0744 n n

log n 0.0744

n 100.0744

t

P r

I Is Ic

F t

r

t Pt

r

tt t

Is = Prt

Is

P

r

t

P = 1, 000, 000

r = 0.25% = 0.0025

t = 1

Is

Is = Prt

Is = (1, 000, 000)(0.0025)(1)

Is = 2, 500

P = 50, 000

r = 10% = 0.10

t =9

12

Is

M t =M

12

Is = Prt

Is = (50, 000)(0.10)

9

12

Is = (50, 000)(0.10)(0.75)

Is = 3, 750

P r t

P =Isrt

=

1, 500

(0.025)(4)P = 15, 000

r =IsPt

=

4, 860

(36, 000)(1.5)r = 0.09 = 9%

t =IsPr

=

275

(250, 000)(0.005)t = 0.22

Is = Prt = (500, 000)(0.125)(10)

Is = 625, 000

r = 7% = 0.07

t = 2

IS = 11, 200

P

P =Isrt

=

11, 200

(0.07)(2)

P = 80, 000

P = 500, 000

Is = 157, 500

t = 3

r

r =IsPt

=

157, 500

(500, 000)(3)

r = 0.105 = 10.5%

P

r = 5% = 0.05

Is =1

2

P = 0.5P

t

t =IsPr

=

0.5P

(P )(0.05)

t = 10

t r

F

F = P + Is

F

P

Is

Is Prt

F = P + Prt

F = P (1 + rt)

F = P (1 + rt)

F

P

r

t

P = 1, 000, 000, r = 0.25% = 0.0025

F

F

Is P F = P + Is.

F = P (1 + rt)

t = 1

Is = Prt

Is = (1, 000, 000)(0.0025)(1)

Is = 2, 500

F = P + Is

F = 1, 000, 000 + 2, 500

F = 1, 002, 500

F

F = P (1 + rt)

F = (1, 000, 000)(1 + 0.0025(1))

F = 1, 002, 500

t = 5

Is = Prt

Is = (1, 000, 000)(0.0025)(5)

Is = 12, 500

F = P + Is

F = 1, 000, 000 + 12, 500

F = 1, 012, 500

F = P (1 + rt)

F = (1, 000, 000)(1 + 0.0025(5))

F = 1, 012, 500

P r t I

P r t I

1

2

I F

I F P F r = 9.5%

I = 9, 500 P = 300, 000 t = 5 I = 16, 250 F = 1, 016, 250

1

4

P

r P r

P (1 + r) = P (1 + r) 100, 000 1.05 = 105, 000P (1 + r)(1 + r) = P (1 + r)2 105, 000, 1.05 = 110, 250P (1 + r)2(1 + r) = P (1 + r)3 110, 250 1.05 = 121, 550.63P (1 + r)3(1 + r) == P (1 + r)4 121, 550.63 1.05 = 127, 628.16

(1 + r) 1 + r

r

F = P (1 + r)t

P

F

r

t

Ic

Ic = F P

P = 10, 000

r = 2% = 0.02

t = 5

F

Ic

F = P (1 + r)t

F = (10, 000)(1 + 0.02)5

F = 11, 040.081

Ic = F PIc = 11, 040.81 10, 000Ic = 1, 040.81

F

P = 50, 000

r = 5% = 0.05

t = 8

F

Ic

F = P (1 + r)t

F = (50, 000)(1 + 0.05)8

F = 73, 872.77

Ic = F PIc = 73, 872.77 50, 000Ic = 23, 872.77

F

P = 10, 000

r = 0.5% = 0.005

t = 12

F

F

F = P (1 + r)t

F = (10, 000)(1 + 0.005)12

F = 10, 616.78

F t r

F = P (1 + r)t

P

P (1 + r)t = F

P (1 + r)t

(1 + r)t=

F

(1 + r)t

P =F

(1 + r)t

P = F (1 + r)t

P

P =F

(1 + r)t= F (1 + r)t

P

F

r

t

F = 50, 000

r = 10% = 0.1

t = 7

P

P

P =F

(1 + r)t

P =50, 000

(1 + 0.1)7

P = 25, 657.91

F = 200, 000

r = 1.1% = 0.011

t = 6

P

P

P =F

(1 + r)t

P =200, 000

(1 + 0.011)6

P = 187, 293.65

P r t Ic

P r t Ic F

Fc = 23, 820.32 Ic = 3, 820.32

Fc = 25, 250.94 Ic = 250.94

Fc = 90, 673.22 Ic = 2, 673.22

P = 89, 632.37

tt

r Ict

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

m i(m)

j

j =i(m)

m=

n

n = tm =

r

i(m) j

r, i(m), j

i(m)

i(1) = 0.02

0.02

1

= 0.02 = 2%

i(2) = 0.02

0.02

2

= 0.01 = 1%

i(3) = 0.02

0.02

4

= 0.005 = 0.5%

i(12) = 0.02

0.02

12

= 0.0016 = 0.16%

i(365) = 0.02

0.02

365

F = P (1 + j)t

j = i(m)

m t mt

F = P

1 +

i(m)

m

!mt

F =

P =

i(m) =

m =

t =

F = P (1 + j)t

F = P

1 +

i(m)

m

!mt

ji(m)

m

t mt

P = 10, 000

i(4) = 0.02

t = 5

m = 4

F

P

j =i(4)

m=

0.02

4

= 0.005

n = mt = (4)(5) = 20 .

F = P (1 + j)n

= (10, 000)(1 + 0.005)20

F = 11, 048.96

Ic = F P = 11, 048.96 10, 000 = 1, 048.96

P = 10, 000

i(12) = 0.02

t = 5

m = 12

F

P

j =i(12)

m=

0.02

12

= 0.0016

n = mt = (12)(5) = 60 .

F = P (1 + j)n

= (10, 000)(1 + 0.0016)60

F = 11, 050.79

Ic = F P = 11, 050.79 10, 000 = 1, 050.79

P = 50, 000

i(12) = 0.12

t = 6

m = 12

F

F = P

1 +

i(12)

m

!tm

F = (50, 000)

1 +

0.12

12

(6)(12)

F = (50, 000)(1.01)72

F = 102, 354.97

P =F

1 +

i(m)

m

!mt

F =

P =

i(m) =

m =

t =

F = 50, 000

t = 4

i(2) = 0.12

P

j =i(2)

m=

0.12

2

= 0.06

n = tm = (4)(2) = 8

P =F

(1 + j)n

P =50, 000

(1 + 0.06)8=

50, 000

(1.06)8= 31, 370.62

F = 25, 000

t = 21

2

i(4) = 0.10

P

j =i(4)

m=

0.10

4

= 0.025

n = tm = (21

2

)(4) = 10

P =F

(1 + j)n

P =25, 000

(1 + 0.025)10=

25, 000

(1.025)10= 19, 529.96

i(m)

F

F

i(m) x =1

j=

m

i(m)m = x(i(m))

m ! 1 x = mi(m)

! 1 m

x =m

i(m)

F = P

1 +

i(m)

m

!mt

F = P

1 +

1

x

xi(m)t

F = P

1 +

1

x

xi(m)t

x

1 +

1

x

x

x ! 11 +

1

x

xe

P i(m)

F t

F = Pei(m)t

P = 20, 000

i(m) = 0.03

t = 6

F

F = Pei(m)t

F = Pei(m)t

= 20, 000e(0.03)(6) = 20, 000e0.18 = 23, 944.35

P = 3, 000

F = 3, 500

i(12) = 0.25% = 0.0025

m = 12

j =i(12)

m=

0.0025

12

t

F = P (1 + j)n

3, 500 = 3, 000

1 +

0.0025

12

n

3, 500

3, 000=

1 +

0.0025

12

n

n

log

3, 500

3, 000

= log

1 +

0.0025

12

n

log(1.166667) = n log

1 +

0.0025

12

n = 740.00

t = nm =740

12

= 61.67

1, 000 300 12%

F = 1, 300

m = 2

i(2) = 0.12

j =i(2)

2

=

0.12

2

= 0.06

n t

F = P (1 + j)n

1, 300 = 1, 000(1 + 0.06)n

1.3 = (1.06)n

log(1.3) = log(1.06)n

log(1.3) = n log(1.06)

n =log 1.3

log(1.06)= 4.503

300

n = 5 t = nm =5

2

= 2.5

1, 000 300

n

n = 5 n = 4.503 t

F = 15, 000

P = 10, 000

t = 10

m = 2

n = mt = (2)(10) = 20

i(2)

F = P (1 + j)n

15, 000 = 10, 000(1 + j)20

15, 000

10, 000= (1 + j)20

1.5 = (1 + j)20

(1.5)120

= 1 + j

(1.5)120 1 = j

j = 0.0205

j =i(m)

m

0.0205 =i(2)

2

i(2) = (0.0205)(2)

i(2) = 0.0410 4.10%

F = 2P

t = 10 s

m = 4

n = mt = (4)(10) = 40

i(4)

F = P (1 + j)n

2P = P (1 + j)n

2 = (1 + i)40

(2)

1/40= 1 + j

(2)

1/40 1 = j

j = 0.0175 1.75%

j =i(4)

m

0.0175 =i(4)

4

i(4) = (0.0175)(4)

i(4) = 0.070 7.00%

i(1)

i(4) = 0.10

m = 4

i(1)

F1

= F2

P (1 + i(1))t = P

1 +

i(4)

m

!mt

P

(1 + i(1))t =

1 +

i(4)

m

!mt

1

t

(1 + i(i)) =

1 +

0.10

4

4

i(i) =

1 +

0.10

4

4

1 = 0.103813 10.38%

j

j = (1.025)4 1

F1

= F2

t t = 1

P t

i(12) = 0.12 m = 12 P t

i(1) = m = 1 P t

F1

F2

F1

= F2

P

1 +

i(1)

1

!(1)t

= P

1 +

i(12)

12

!12t

1 +

i(1)

1

!=

1 +

0.12

12

12

i(1) = (1.01)12 1

i(1) = 0.126825%

i(2) = 0.08 m = 2 P t

i(4) = m = 4 P t

F1

F2

F1

= F2

P

1 +

i(4)

4

!(4)t

= P

1 +

i(2)

2

!(2)t

1 +

i(4)

4

!4

=

1 +

0.08

2

2

1 +

i(4)

4

!4

= (1.04)2

1 +

i(4)

4

=