# Precalculus: An Investigation of Functions

David Lippman, Pierce College

Melonie Rasmussen, Pierce College

Copyright Year: 2017

Publisher: David Lippman and Melonie Rasmussen

Language: English

## Formats Available

## Conditions of Use

Attribution-ShareAlike

CC BY-SA

## Reviews

The book covers all the topics typically covered in a Pre-Calculus class. Additionally, the book contains sufficient material for a trigonometry course. There is no review of algebra. Index is very accurate and answers to selected exercises are... read more

The book covers all the topics typically covered in a Pre-Calculus class. Additionally, the book contains sufficient material for a trigonometry course. There is no review of algebra. Index is very accurate and answers to selected exercises are helpful to students. Front Matter is well written.

Overall, I found the content to be accurate. Definitions could’ve been stated more precisely then explained in words. Some properties need more clarity, for example when the inverse properties for “logarithms” were introduced on pages 277 (and on 289), restriction on “x” should’ve been included so that students can be reminded of the domain of the composition.

Conceptually is good. Updating applications can be done with ease.

Overall, the book was clear, and it was written with the students in mind. Many definitions were informal. Trigonometry section did have a very good introduction and good explanations.

This book is consistent throughout. The authors ensured that a motivation problem is introduced prior to stating definitions. Then the author introduced the necessary definition, some in an informal way, with good examples.

This book follows the standard set up of any Pre-Calculus course. I found devoting a whole chapter on Linear Functions to be a refreshing concept.

The logical flow of topics in this book is the standard flow of topics in any Pre-Calculus book. I did like the learning objectives at the beginning (or within) of each section, such objectives will help students with the mathematical language.

I did not experience any interface issues with this book.

I did not see any grammatical errors in the text.

The book is culturally neutral.

This book is informal, intuitive, and straightforward. It was written with students in mind. I would like to see a review of basic algebra. If the instructor is careful to address the places where the book is too informal, I think this book will be a fine Precalculus textbook.

The topics were well addressed, and in particular, the Trigonometry concepts from Chapter 5 onwards were well traversed. read more

The topics were well addressed, and in particular, the Trigonometry concepts from Chapter 5 onwards were well traversed.

As such no errors were seen.

Since the books teaches basic concepts with great detail, updates if and when required can be made to questions dealing with applications, in a relatively simple manner. However, conceptually there will be no issues.

All terminologies and allusions to variables were used in as clear of a form as possible.

The textbook was consistent throughout.

I had to especially used Chapters 5-8 from this textbook. I found that there was no problem with the format, and each Chapter could be assigned/worked on individually as long as basic concepts were clear.

Very logically and well distributed topics and content. The 'Try it now' questions were very useful, both for me as an instructor as well as for the students.

No issues were found, and students were able to navigate through the content well.

No grammatical errors.

No such issues were found.

As an instructor teaching Trigonometry for the first time, I found the textbook extremely useful, and would definitely recommend it to other professors.

This book is excellent when it comes to providing students with real-world examples for the topics, but teachers need to be careful when using this book that they define the vocabulary exactly the same as the book does, so as not to confuse the... read more

This book is excellent when it comes to providing students with real-world examples for the topics, but teachers need to be careful when using this book that they define the vocabulary exactly the same as the book does, so as not to confuse the students. . . For example:

Book's definition of a FUNCTION: "A rule for a relationship between an input, or independent, quantity and an output, or dependent, quantity in which each input value determines one output value."

How some teachers define a FUNCTION: "A relationship that shows for every x-value there is one and only one y-value."

These definitions are both saying the same thing, but for a student who is learning it for the first time, such a drastic difference in wording could cause confusion. Here is another example:

Book's definition of RATE OF CHANGE: "Describes how the output quantity changes in relation to the input quantity."

How some teachers define RATE OF CHANGE: "How fast or how slow a function is increasing or decreasing." . . . Be Careful!

I have used this textbook as an instructor and also as a tutor for many years - I have found no evidence of errors or biased content.

This textbook is very well written and the content is arranged in a way that is most conducive to learning. Within each Section of this book, new vocabulary is defined among several example problems, there is a "List of Important Topics" at the end, and the number of exercises for the students is (for the most part) less than 50 problems so as to make assigning homework easier and less overwhelming / time-consuming.

Also, what makes this textbook somewhat unique and extremely helpful is that it writes out the solutions step-by-step for each of the example problems, and it warns students along the way to "resist the urge" to make common mistakes when solving.

With further regards to relevance, as a teacher, I found it beneficial for my students to teach the entire content of this textbook with the exception of 2 Sections: (5.2) "Linear and Angular Velocity" (skipped because I felt it got too deep into Physics) and (8.5) "Parametric equations; Lissajous figures" (skipped because I felt it to be a bit heavy for Pre-Calculus - better saved for Calculus 1 & 2)

Some of the definitions for the vocabulary are a bit "wordy" in my opinion, but if students read through them carefully they all make sense and even help answer the ever-popular question, "How am I ever going to use this in the real world?!"

There is just the right amount of Geometry review before the chapters on Trigonometric functions (sine, cosine, etc.), and when students learn how to graph these functions, the book gives the practical example of using these graphs to monitor the changing population of Elk. . . There is never any mention of the acronym, "S-O-H-C-A-H-T-O-A," however, which I highly recommend as I have found to be very useful in helping students understand this section.

The terminology and topics throughout this textbook are designed in a very consistent framework - each section introduces the material using real-world application examples, shows further examples of how to solve problems step-by-step, introduces new vocabulary where appropriate, gives students a list of topics to review, and then allows for a moderate amount of independent practice.

Each chapter is divided into several sections, which help break apart the overall topic and allows the teacher to present the material in a way that is neither too fast nor too slow for students to understand. There is even a separate section for allowing students to practice writing Inequalities using Set-Builder and Interval notation - something I've seen other textbooks leave out.

Very organized content throughout; Easy to find specific topics and/or example problems.

Many charts, tables, and graphs are used to teach the different mathematical concepts throughout this textbook. I have found them to be essential to the explanations and overall understanding of the topics. Students, in my own experience, have not encountered any issues with how to interpret these images.

I have not found there to be any grammatical errors. . . although I am a Math instructor, not an English Language instructor! ;)

No offensive or biased references to race, culture, or ethnicity are found among the real-world example problems in this textbook.

HIGHLY RECOMMEND!!

Excellent resources for students who have completed Algebra 2, and before entering Calculus 1.

This textbook covers all areas of a Pre-Calculus Functions book. It includes everything you would need for at least two terms, one of an Intro to Functions and one of Trigonometry. The book is written in an easy to understand format compared to... read more

This textbook covers all areas of a Pre-Calculus Functions book. It includes everything you would need for at least two terms, one of an Intro to Functions and one of Trigonometry. The book is written in an easy to understand format compared to harder texts. There is no review of algebra but the easy to understand textbook with key items in bold and blocked out help.

I have not found any errors on any of the problems I have worked. The appears to be no bias on any of the problems.

The problems introduces in this textbook are current and relevant to society today. I don't see them needing to be updated or changed in the near future. Students are able to see the relevance for learning the functions.

This textbook is easy to follow and understand. Any mathematical notation used is explained so that even a "non-math" person can follow. Students should be able to understand the text on their own with little help from an instructor. Terminology is defined also.

The nice thing about this textbook is that the book follows the same structure for each function defined and talked about in the text. Students are shown problems and answers are given in four methods; verbally, numerically, algebraically, and graphically. So no matter the learning style of the student they have it presented in multiple ways.

This textbook is set up very nicely. The first four chapters lends itself very nicely to a Functions course. the second half of the text will satisfy and Trigonometry class. The way it is broken up, instructors can use the book in any order.

The flow of this book is in a logical manor. The topics appear in an order which will aid the students. Each section of the book has problems worked out for the student and then problems for the student to try along the way. This aids the student to scaffold the information as they go along.

There appeared to be no issues with the navigation of this text book. Links within the book worked seamlessly. I would have liked to see it set up so that it looked like a book with two pages at a time. (That may just have been my browser)

I saw no grammatical errors.

There was no culturally insensitive or offensive material.

This book is good introductory book to Functions and Trigonometry. Its ease of reading won't scare students away from the material. It would have been nice though to have a quick algebra refresher. Overall I found it to be well written and set up for easy adoption in a class.

This text covers all areas of a standard Precalculus class, with or without trigonometry. There is no algebra review. There is an effective index and there are solutions to selected exercises in the back. There is no separate glossary, but key... read more

This text covers all areas of a standard Precalculus class, with or without trigonometry. There is no algebra review. There is an effective index and there are solutions to selected exercises in the back. There is no separate glossary, but key terms are in bold and defined within each section.

All of the problems that I worked, I found to be error-free and unbiased.

All of the applications problems that I read would maintain their relevancy. This text would not need to be updated regularly.

This text is written with excellent clarity. All symbolic notation is clearly explained. This is a text that students would be able to read and follow on their own. Formal notation and terminology is used and are explained very well.

The book is very consistent in terms of how each function is presented. First, an application is given of how the function is used in modeling. This gives the students an idea of the purpose of each function. Then symbolic notation is introduced for the function, followed by an in-depth study of graphs and modeling. In the Preface, the authors state that the rule of four is used. The functions are all addressed verbally, graphically, numerically, and algebraically. I found this to be true as I reviewed each one.

Each section is an appropriate length. Examples and definitions are clearly labeled. The text could easily be used for a Precalculus without trigonometry class and a Precalculus with trigonometry class. No algebra review is given, nor are there any appendices on algebra topics.

The organization, structure, and flow of this text is very logical and clear.

This text is easy to navigate. The table of contents is useful and the page headers are as well. However, I did find that the symbol for the set of real numbers did not get printed on page 23. This is the only distortion/display issue that I came across.

This text contains no grammatical errors that I found.

The examples and exercises in this book apply the mathematics to a wide variety of scenarios. None of them appear to be culturally insensitive or offensive.

The only thing I would like to see added to this text would be a basic algebra review, or some appendices on algebra that students could use to brush up on pre-requisite skills if necessary. I also would like to see an introduction to the difference quotient. Otherwise, I thought this was an excellent Precalculus text.

This book is very thorough in the topics that are covered and seems to contain nearly everything that I would expect to find in a precalculus course. The only topics that I wish were included is finding oblique asymptotes and solving for complex... read more

This book is very thorough in the topics that are covered and seems to contain nearly everything that I would expect to find in a precalculus course. The only topics that I wish were included is finding oblique asymptotes and solving for complex zeroes, however I think that it would be very simple to add a supplementary resource for those topics.

The book seems to be very accurate and I did not find any errors while examining the text.

In my opinion the book has current applications and will remain relevant for a long time. I did find one link for source material for a problem that was no longer valid, but this could be easily rectified in a future update.

This text was written very clearly using correct terminology. This is a good source for students to use in learning precalculus, but due to that fact that many students have difficulty reading math texts they may have to read a section more than once to fully grasp a concept.

The book has a very consistent lay out and even color codes different features. For example, blue headings are for concepts and definitions, grey headings are for worked out examples and green headings are problems for students to try.

The book has chapters that focus on broad topics and each chapter is subdivided into more specific concepts allowing for the instructor to easily organize for specific lessons.

The organization and order for the presentation of material seems to be in a logical order. What I really enjoy is that within each section the text has worked out examples for the student to learn from and then has a similar problem for the student to try with answers to the try it problems at the end of the section.

When I went through the book all of the images were clear and I did not have any navigational issues. I do recommend viewing the book in a two page book layout because the margins are set up for it to be a book. If you do not view it in a two page book layout it does not negatively effect anything, the reader will just notice that the margins change every page.

I did not find any grammatical errors within the text.

I did not find anything within the book that I would view to be culturally insensitive or offensive.

I think that this is a really well written book that has some really interesting practice problems and applications. One thing that I really appreciated was the fact that the book had links for resources that could be used with it. For example, there was a link to a web-based linear regression calculator for students to use. This is definitely a book that I will consider adoption for my future precalculus courses.

The book covered all the topics that any instructor might want to cover in one or two-semester long in a college level Pre-Calculus course. It has 12 chapters and is divided into 3 groups. First group is the first 4 chapters that cover functions... read more

The book covered all the topics that any instructor might want to cover in one or two-semester long in a college level Pre-Calculus course. It has 12 chapters and is divided into 3 groups. First group is the first 4 chapters that cover functions creating a solid foundation for the course. Second is all about Trigonometry and its applications. I appreciate the way it began this section using angles and unit circles and then moving on to triangles. Third piece is dealt with the advanced topics with systems and equations, a little of discrete math, analytic geometry and a prep for calculus. Interestingly, this is exactly how I teach the course and divide the semester with the content of the course. I loved the figures and illustrations that used contemporary styles with color contrast and labeling.

It seems like authors have taken a great effort to ensure the validity and accuracy of the content in the book. Each chapter’s manuscript underwent at least two rounds of review and revision by a panel of active Precalculus instructors. Another team of experts checked all text, examples, and graphics for mathematical accuracy. A third team of experts tested the accuracy of the Answer Key, by re-working every solution. The Solutions Manual was rigorously checked for accuracy.

The content of the book seems to be very contemporary and up-to-date. It used very relevant real-life problems and examples. It added media buttons that can be changed or updated if necessary. At the end of every chapter the access to online resources are very useful.

The authors used clear concise English language without any redundancy. It clearly says what it is planning to do every step and showed detailed work in the examples. It also added analysis that musty be very helpful to the students. The graphs shown are just like I would do in class on the board. All the explanations used “we” that is very appropriate for the students to see the line of thinking within the mind of the author.

The format is very consistent with mathematical languages and terminology. It appears that the authors have used their own text repeatedly and they have done a good job providing links to the online sources. The figure for vertical and horizontal reflection is beautifully done. This is true for all mathematical concepts. The figures are very valuable learning component for the students to see and are consistent with the topics. It used very appropriate standard mathematical terminology throughout.

The book used standard sequence of topics. It can easily be divided in two semester long courses if needed. In some schools advanced topics are not covered in PreCalculus courses and hence these topics can be avoided without any trouble.

The organization/structure/flow of this book is outstanding with learning objectives at the beginning of every section. This helps the instructors to clearly focus on the topics and techniques.

I did not experience any interface issues with this text. The links within the text, takes the reader to statements of previous theorems, previous examples, particular exercises, a referenced diagram, definition, or solution of an exercise, which is a handy feature. At the end of every chapter the key concepts quickly links to the topic discussed in the sections.

I did not see any grammatical errors in the text.

The book seemed to be culturally neutral. I did not notice any race or religion based problems. The problems used “you” or “your” many times. Other problems centered on the mathematical concepts.

It was fun-time reading this book. I thoroughly enjoyed it and may be I will even adopt it in the fall of 2015.

## Table of Contents

- Chapter 1: Functions
- Chapter 2: Linear Functions
- Chapter 3: Polynomial and Rational Functions
- Chapter 4: Exponential and Logarithmic Functions
- Chapter 5: Trigonometric Functions of Angles
- Chapter 6: Periodic Functions
- Chapter 7: Trigonometric Equations and Identities
- Chapter 8: Further Applications of Trigonometry

## Ancillary Material

## About the Book

*Precalculus: An Investigation of Functions* is a free, open textbook covering a two-quarter pre-calculus sequence including trigonometry. The first portion of the book is an investigation of functions, exploring the graphical behavior of, interpretation of, and solutions to problems involving linear, polynomial, rational, exponential, and logarithmic functions. An emphasis is placed on modeling and interpretation, as well as the important characteristics needed in calculus.

The second portion of the book introduces trigonometry. Trig is introduced through an integrated circle/triangle approach. Identities are introduced in the first chapter, and revisited throughout. Likewise, solving is introduced in the second chapter and revisited more extensively in the third chapter. As with the first part of the book, an emphasis is placed on motivating the concepts and on modeling and interpretation.

In addition to the paper homework sets, algorithmetically generated online homework is available as part of a complete course shell package, which also includes a sample syllabus, teacher notes with lecture examples, sample quizzes and exams, printable classwork sheets and handouts, and chapter review problems. If you teach in Washington State, you can find the course shell in the WAMAP.org template course list. For those located elsewhere, you can access the course shell at MyOpenMath.com. A self-study version of the online course exercises is also available on MyOpenMath.com for students wanting to learn the material on their own, or who need a refresher.

## About the Contributors

### Authors

**David Lippman** received his master’s degree in mathematics from Western Washington University and has been teaching at Pierce College since Fall 2000.

**Melonie Rasmussen** received her master’s degree in mathematics from Western Washington University and has been teaching at Pierce College since Fall 2002. Prior to this Melonie taught for the Puyallup School district for 6 years after receiving her teaching credentials from Pacific Lutheran University.