Foreword to the Student

Assumed Prerequisite Knowledge

This course is designed for the college student who has successfully completed Arithmetic, Prealgebra, Elementary Algebra, Intermediate Algebra, and Trigonometry. While mastery of all of these topics is highly predictive of success in Precalculus, it's understandable if you forgot a handful of things from each of these courses. There is some review embedded within this text if the topic is something that came from your most recent course (which was likely Trigonometry, but could have been Intermediate Algebra); however, the instruction is more terse for these topics and assumes that you just need a reminder. If you need a more thorough instruction of the prerequisite material, I suggest you crack open your old Algebra textbooks.

Pace

This course covers between 50 and 55 sections in a single semester. If your class meets twice per week, you can assume that you will be going through approximately two sections per day. If your class meets three days per week, you will cover roughly one-and-a-half sections per class session.¹ The point of these statements is that a Precalculus course is packed with material. It covers in one semester what most high school math courses cover in one to two full years. You must keep this in mind when planning your schedule. For each hour of in-class (or online) lecture, you can assume you will spend two hours outside of class reading, conceptualizing, and doing homework.

Assumed Mastery (i.e., Topics That Will Not Be Reviewed)

As opposed to the "assumed prerequisite knowledge" mentioned earlier, there are some things that are so low-level that we don't have time to review them in this text. When writing a textbook for Mathematics, the author must make a few realistic assumptions about the skill level of the reader. In this course, it is assumed the reader has mastery² of the following skills, concepts, definitions, and theorems.

The reader should be able to...

• use arithmetic operations including addition, subtraction, multiplication, division, simple roots, and simple powers,
• perform all of the operations listed above on real numbers (yes, including fractions) without the aid of technology (i.e., a calculator),
• use the fundamental axioms of arithmetic, which are
  • the Commutative Properties of Addition and Multiplication,
  • the Associative Properties of Addition and Multiplication,
  • the Additive and Multiplicative Identities, and
  • the Distributive Properties
• understand like and unlike terms and the arithmetic involved in combining like terms,
• properly use the Order of Operations,
• order numbers (e.g., knowing that \( -\pi \gt -3 \) and \( \frac{3}{4} \lt \frac{7}{8} \)) and understand where these fit on the number line,
• simplify compound rational expressions (a.k.a., compound fractions or complex fractions),
• perform substitution (into expressions and functions),
• know when and how to apply operations (e.g., addition, powers, etc.) to both sides of an equation simultaneously,
• solve linear equations,
• solve linear inequalities,
• perform basic polynomial arithmetic (i.e., add, subtraction, multiply, and divide polynomials),
• simplify radical expressions,
• compute an average,
• factor out the GCF from an expression,
• factor trinomials (using a method other than factoring by grouping),
• factor binomials (using the difference of squares or sum/difference of cubes formulas),
• solve an equation involving several variables for one of the variables,
• apply the Laws of Exponents when simplifying expressions, 
• understand the subtleties of finding the domain of square root functions and rational functions,
• graph the base graphs for \( y = x \), \( y = x^2 \), \( y = x^3 \), \( y = \sqrt{x} \), \( y = \sqrt[3]{x} \), \( y = |x| \), \( y = \frac{1}{x} \), and \( y = \frac{1}{x^2} \), and
• use the Zero Factor Property to solve quadratic and polynomial equations.

The reader is also assumed to have total understanding (without any need for exposition) of...

• the magnitude or "size" of a number,
• infinity (from a conceptual, not rigorous, point-of-view),
• the difference between an expression and an equation,
• the fact that \( |x^2| = x^2 \) (i.e., the absolute value of a non-negative number is the number itself),
• the possibility of introducing extraneous solutions when applying even powers to both sides of an equation (therefore, always check answers when raising both sides of an equation to an even power), and 
• the difference between an implication, \( \implies \), and an equality, \( = \).

Furthermore, the reader is assumed to know (without need for exposition) the definitions of...

• average,
• dividend,
• equality,
• equation,
• exclusive (with regard to endpoints of intervals),
• expression,
• finite number,
• horizontal,
• inclusive (with regard to endpoints of intervals),
• inequality (and the inequality symbols),
• line segment,
• number line,
• radical,
• radicand,
• rational,
• strict inequality,
• triangle,
• variable, and
• vertical.

Finally, it is assumed the reader has complete understanding (without any need for review) of the following theorems:

• Pythagorean Theorem
• Extraction of Roots. Since this is known by other names, I will be explicit here - you need to know that \( x^2 = k \implies x = \pm \sqrt{k} \).
• Quadratic Formula
• Laws of Exponents
• Zero Factor Property