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Introduction
Pre-calculus is a rather difficult topic to define or describe. There’s a little bit of this, a lot of that, and a smattering of something else. But you need the mathematics considered to be pre-calculus to proceed to what changed me into a math major: calculus! Yes, believe it or not, I started out as a biology major — inspired by my high school biology teacher. Then I got to the semester where I was taking invertebrate zoology, chemistry, and calculus. (Yes, all at the same time.) All of a sudden, there was a bright light! An awakening! “So this is what mathematics can be!” Haven’t turned back since. Calculus did it for me, and my great preparation for calculus made the adventure wonderful.
Pre-calculus contains a lot of algebra, some trigonometry, some geometry, and some analytic geometry. These topics all get tied together, mixed up, and realigned until out pops the mathematics you’ll use when working with calculus. I keep telling my calculus students that “calculus is 60 percent algebra.” Maybe my figures are off a bit, but believe me, you can’t succeed in calculus without a good background in algebra (and trigonometry). The geometry is very helpful, too.
Why would you do 1,001 pre-calculus problems? Because practice makes perfect. Unlike other subjects where you can just read or listen and absorb the information sufficiently, mathematics takes practice. The only way to figure out how the different algebraic and trigonometric rules work and interact with one another, or how measurements in degrees and radians fit into the big picture, is to get into the problems — get your hands dirty, so to speak. Many problems given here may appear to be the same on the surface, but different aspects and challenges have been inserted to make them unique. The concepts become more set in your mind when you work with the problems and have your solutions confirm the properties.
What You’ll Find
This book contains 1,001 pre-calculus problems, their answers, and complete solutions to each. There are 16 problem chapters, and each chapter has many different sets of questions. The sets of questions are sometimes in a logical, sequential order, going from one part of a topic to the next and then to the next. Or sometimes the sets of questions represent the different ways a topic can be presented. In any case, you’ll get instructions on doing the problems. And sometimes you’ll get a particular formula or format to use. Feel free to refer to other mathematics books, such as Yang Kuang and Elleyne Kase’s Pre-Calculus For Dummies, my Algebra II For Dummies, or my Trigonometry For Dummies (all published by Wiley) for even more ideas on how to solve some of the problems.
Instead of just having answers to the problems, you’ll find a worked-out solution for each and every one. Flip to the last chapter of this book for the step-by-step processes needed to solve the problems. The solutions include verbal explanations inserted in the work where necessary. Sometimes, the explanation may offer an alternative procedure. Not everyone does algebra and trigonometry problems exactly the same way, but this book tries to provide the most understandable and success-promoting process to use when solving the problems presented.
How This Workbook Is Organized
This workbook is divided into two main parts: questions and answers. But you probably figured that out already.
Part 1: The Questions
The chapters containing the questions cover many different topics:
Review of basic algebraic processes:Chapters 1 and 2 contain problems on basic algebraic rules and formulas, solving many types of equations and inequalities, and interpreting and using very specific mathematical notation correctly. They thoroughly cover functions and function properties, with a segue into trigonometric functions.
Graphing functions and transformations of functions: Functions and properties of functions are a big part of pre-calculus and calculus. You work with operations on functions, including compositions. These operations translate into transformations. And all this comes together when you look at the graphs of the functions. Transformations of functions help you see the similarities and differences in basic mathematical models — and the practice problems help you see how all this can save you a lot of time in the end.
Polynomial functions: Some of the more familiar algebraic functions are the polynomials. The graphs of polynomials are smooth, rolling curves. Their characteristics include where they cross the axes and where they make their turns from moving upward to moving downward or vice versa. You get to practice your equation-solving techniques when determining the x-intercepts and y-intercept of polynomial functions.
Exponential and logarithmic functions: You’re not in Kansas anymore, so it’s time to leave the world of algebraic functions and open your eyes to other types: exponential and logarithmic, to name two. You practice the operations specific to these types of functions and see how one is the inverse of the other. The applications of these functions are closer to real-world than most others in earlier chapters.
Trigonometric functions: Trigonometric functions take being different one step further. You’ll see how the input values for these functions have to be angle measures, not just any old numbers. The trig functions have their own rules, too, and lots of ways to interact, called identities. Solving trig identities helps you prepare for that most exciting process in calculus, where you get to find the area under a trigonometric curve. So, keep your eye on that goal! And the trig applications are some of my favorite — all so easy to picture (and draw) and to solve.
Complex numbers and polar coordinates: Complex numbers were created; no, they aren’t real or natural. Mathematicians needed to solve problems whose solutions were the square roots of negative numbers, so they adopted the imaginary number i to accomplish this task. Performing operations on complex numbers and finding complex solutions are a part of this general arena. Polar coordinates are a way of graphing curves by using angle measures and radii. You open up a whole new world of curves when you practice with these problems dealing with polar graphs.
Conic sections: A big family of curves belongs in the classification of conics. You find the similarities and differences between circles, ellipses, hyperbolas, and parabolas. Exercises have you write the standard forms of the equations so
