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Precalculus: An Investigation of Functions

David Lippman and Melonie Rasmussen
Publication Date: 
Number of Pages: 
Electronic Book
open source
[Reviewed by
Mike Kenyon
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At Green River Community College, we began using Lippman/Rasmussen for our two-quarter precalculus sequence in the fall of 2011. This was a departure from our usual procedure. Normally we phase the new text in so that students who began the sequence with the previous text have an opportunity to complete it without having to buy a new book. In this case, the low cost for a printed copy and the fact that it was available for free online made us decide that the inconvenience for some students was outweighed by the substantial savings for others.

We considered about two dozen texts in our adoption process; the others were all from traditional publishers. The Lippman/Rasmussen text was one of three that emerged as finalists. Price did not become a significant consideration until we had decided on finalists; up to that point, we were simply interested in finding the highest-quality finalists available. Once the finalists were selected, it became clear that the Lippman/Rasmussen text was equal or superior to the others in quality and far outpaced them in cost. The vote of our full-time faculty was, in fact, unanimous.

The text had a positive effect on the classroom instructional atmosphere from the very beginning. Many students came to class on the first day with a positive attitude borne of having been to the bookstore and found that their textbook would cost $20 rather than over $100, and even spending that much was optional. Moreover, the vast majority of students had the textbook in one form or another from the outset and so didn’t face the prospect of falling behind because they couldn’t get it until a financial aid check came in. Those few students who said, after a couple of days, “I couldn’t get the book yet” effectively identified themselves as needing a little extra guidance in how to be a successful in a collegiate mathematics course — and, in most cases, got that attention quickly and ended up going in the right direction rather than languishing for a longer time.

The text divides nicely into two pieces. The first four chapters correspond to our Precalculus I; Chapters 5–8, with an emphasis on trigonometry, go with our Precalculus II. Not surprisingly, the correspondence is not perfect; it has been a very long time since we had a precalculus book that we did not believe required supplementation in some area(s). In this case, we needed to add some material on vectors and conic sections. The difference is the ease with which this text can be supplemented. In fact, the authors encourage that, suggesting that we send them our supplementary materials to include, or that we could create a version of the text that is specific to our college. Thus far, we have not chosen to do that; short handouts and worksheets do the job more than adequately. Students seem more receptive to changes, supplementary handouts, and the like as well, likely because they’re more willing to be flexible with something that’s so much less expensive.

Individual instructors have found certain topics lacking (for example, the treatment of inverse functions from a graphical perspective). This, too, is easily remedied with supplements, and is no different what happens with from conventional texts. The authors have been able to be more responsive more quickly than traditional authors, often correcting errors in the online edition the same day they are identified. Students seem to be less frustrated by errors than when they occur in other texts, possibly because of the lower price. In fact, students take some pride in finding them (especially in the answers to homework problems). We have taken to telling our students that when they find something that seems wrong in the printed edition, they should check the online edition to see if it has been fixed. The authors have recently completed revisions that incorporate suggestions, corrections, and the like while not constituting a new edition — all homework exercises and page numbers are the same, for example, so students using earlier versions will not find the differences problematic.

The first half of the book is rich in applications while still being robust from an algebraic and computational standpoint. In the preface, the authors note that “There is nothing we hate more than a chapter on exponential equations that begins ‘Exponential functions are functions that have the form…” Indeed, each new family of functions is introduced with examples that motivate the need for such a family. Chapter 4, on exponential and logarithmic functions, begins with short descriptions of population growth and financial scenarios. It then uses them to develop exponential models, which are then used to construct tables of values. Those tables, in turn, are used to construct compare-and-contrast graphs with linear and exponential functions. Homework sets also include in-depth, challenging applications (some of which are remixed, with permission, from Precalculus by D.H. Collingwood and K.D. Prince); these require students to form and implement a plan and to work through multiple steps to reach a solution.

The second half of the book is more disappointing. Several sections in Chapters 5–8 do not have a single real-world application in the homework sets. Some of the problems are challenging and help students develop their abilities to solve non-routine, multi-step problems, and students will certainly become computationally proficient (which is important!). But many students will be no closer to understanding why they should care about, for example, working with trig identities. This is perhaps the more disappointing since it comes on the heels of such well-done homework sets in the first four chapters.

Numerous supplements are available at no charge at and These include, among many other features, a day-by-day course guide, discussion forums, algorithmically generated free-response online homework for each section, sample quizzes and exams, and supplemental videos.

The free online book poses some new challenges. Given the cost of printer paper and ink, it’s cheaper for students to buy a copy than to print the pages themselves, but our students have a big enough printing allocation in the campus computer labs that some of them choose to print their own anyway; this is inefficient and, if it becomes commonplace, will require us to make adjustments to that allocation or how it can be used. Moreover, students who use the electronic version now have a good reason to have laptops, iPads, cell phones, and the like in use during class. A large majority of such use that we’ve observed has been on topic, so this seems to be less of a concern than some of us may have feared.

Last spring (Spring Quarter 2012) I taught two first-quarter calculus classes that included students who were the first to have taken both quarters of precalculus from this text. I was largely pleased with my students’ competence with the prerequisite material, and in particular, they were proficient with what Lippman and Rasmussen call the “toolkit” functions, the basic representatives of families of functions (e.g., linear, quadratic, exponential, logarithmic, and so forth; they even used the term “toolkit” routinely, suggesting that they had made enough use of the text to be familiar with some of its unique language). That strong grasp made moving into the various families of functions in calculus go that much more smoothly.

My calculus students were also well versed in looking at problems from multiple perspectives. They used appropriate mathematical language to describe the behavior of functions and were rarely concerned if the function were shown as a graph or even a table rather than a formula, or vice versa. Clearly they were used to going from one form to another, depending on what might be most useful for a given situation; examples and problems like those described above from Chapter 4 are likely a significant contributor.

On the whole, then, we have been very pleased with our move to this text. In fact, we routinely encourage our colleagues to consider Lippman/Rasmussen when they adopt a text. While the second half may not be quite as strong as the first, it compares favorably on quality to conventional texts at, of course, a small fraction of the cost to students. The authors’ responsiveness to corrections and suggestions has made for a flexibility we have rarely encountered, and the minor changes they have made as a result make the text that much better. Most importantly, students engage with the text and carry its ideas effectively into their future courses.

The free version of this book is available from A printed version can be ordered for $16 from

Mike Kenyon teaches at Green River Community College in Auburn, WA.

Front Matter PDF DOC
Chapter 1: Functions
  • 1.1 Functions and Function Notation
  • 1.2 Domain and Range
  • 1.3 Rates of Change and Behavior of Graphs
  • 1.4 Composition of Functions
  • 1.5 Transformation of Functions
  • 1.6 Inverse Functions
Chapter 2: Linear Functions
  • 2.1 Linear Functions
  • 2.2 Graphs of Linear Functions
  • 2.3 Modeling with Linear Functions
  • 2.4 Fitting Linear Models to Data
  • 2.5 Absolute Value Functions
Chapter 3: Polynomial and Rational Functions
  • 3.1 Power Functions & Polynomial Functions
  • 3.2 Quadratic Functions
  • 3.3 Graphs of Polynomial Functions
  • 3.4 Rational Functions
  • 3.5 Inverses and Radical Functions
Chapter 4: Exponential and Logarithmic Functions
  • 4.1 Exponential Functions
  • 4.2 Graphs of Exponential Functions
  • 4.3 Logarithmic Functions
  • 4.4 Logarithmic Properties
  • 4.5 Graphs of Logarithmic Functions
  • 4.6 Exponential and Logarithmic Models
  • 4.7 Fitting Exponentials to Data
Chapter 5: Trigonometric Functions of Angles
  • 5.1 Circles
  • 5.2 Angles
  • 5.3 Points on Circles using Sine and Cosine
  • 5.4 The Other Trigonometric Functions
  • 5.5 Right Triangle Trigonometry
Chapter 6: Periodic Functions
  • 6.1 Sinusoidal Graphs
  • 6.2 Graphs of the Other Trig Functions
  • 6.3 Inverse Trig Functions
  • 6.4 Solving Trig Equations
  • 6.5 Modeling with Trigonometric Equations
Chapter 7: Trigonometric Equations and Identities
  • 7.1 Solving Trigonometric Equations with Identities
  • 7.2 Addition and Subtraction Identities
  • 7.3 Double Angle Identities
  • 7.4 Modeling Changing Amplitude and Midline
Chapter 8: Further Applications of Trigonometry
  • 8.1 Non-right Triangles: Law of Sines and Cosines
  • 8.2 Polar Coordinates
  • 8.3 Polar Form of Complex Numbers
  • 8.4 Vectors
  • 8.5 Parametric Equations
Answers to Selected Exercises PDF DOC