Review of Precalculus

Why Precalculus Matters for Calculus

Calculus builds directly on algebra and trigonometry. When you compute a derivative, you use algebraic manipulation. When you evaluate limits, you use properties of inequalities and absolute values. When you integrate trigonometric functions, you need to know the identities cold. A gap in any of these areas will slow you down at exactly the wrong moment.

This chapter gives you a single, consolidated reference for everything you need. Work through it actively: try the examples before reading the solutions, and return to specific sections whenever a later chapter demands a skill you feel uncertain about.

Sections in This Chapter

Section	Title	What You Will Learn
1.1	Elementary Algebra	Algebraic identities, the binomial theorem, completing the square, quadratic formula
1.2	Inequalities	Rules for manipulating inequalities, including multiplying by negative numbers
1.3	Absolute Value	Definition, properties, and solving equations and inequalities involving absolute value
1.4	Intervals	Open, closed, and half-open intervals; infinite intervals; interval notation
1.5	Neighborhoods	Delta-neighborhoods and deleted neighborhoods, essential for limits
1.6	Straight Lines	Slope, point-slope form, parallel and perpendicular lines
1.7	Exponents and Logarithms	Laws of exponents, natural logarithm, change-of-base formula, laws of logarithms
1.8	Sigma Notation	Summation notation, index of summation, upper and lower limits
1.9	Functions	Domain, range, co-domain, graph of a function, natural domains of common functions
1.10	Piecewise-Defined Functions	Functions defined by different formulas on different intervals
1.11	Even and Odd Functions	Symmetry about the $y$-axis versus symmetry about the origin
1.12	Increasing and Decreasing Functions	Formal definitions and graphical interpretation
1.13	Composition of Functions	The $f\circ g$ operation, domain restrictions, order matters
1.14	One-to-One Functions	Definition, horizontal line test
1.15	Inverse Functions	Construction of $f^{-1}$, graphs as reflections in $y=x$, domain-range swap
1.16	Polynomials, Remainder and Factor Theorems	Polynomial division, remainder theorem, factor theorem
1.17	A Quick Review of Trigonometry	Radian measure, right-triangle and unit-circle definitions, key identities
1.18	Periodic Functions and Trigonometric Graphs	Period, fundamental period, graphs of sin, cos, tan
1.19	Inverse Trigonometric Functions	arcsin, arccos, arctan: domains, ranges, cancellation equations
1.20	Two Important Inequalities	$-

Real-World Connections

Precalculus concepts show up throughout science and engineering:

• Exponential and logarithmic functions model population growth, radioactive decay, compound interest, and signal attenuation.
• Trigonometric functions describe oscillations in electrical circuits, sound waves, mechanical vibrations, and planetary motion.
• Inequalities and absolute values arise in error analysis, tolerances in manufacturing, and epsilon-delta proofs.
• Function composition is the mathematical foundation of how complex systems are built from simpler subsystems.
• Sigma notation is the language of numerical methods, statistics, and discrete mathematics.

Frequently Asked Questions

Do I need to memorize all these formulas?

The most important ones to have truly memorized are: the quadratic formula, the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, the double-angle formulas, and the addition formulas for sine and cosine. The laws of exponents and logarithms should be automatic. Everything else can be derived quickly if you understand where it comes from, so focus on understanding rather than rote memorization.

What if my algebra is weak? Should I study a separate algebra textbook first?

Not necessarily. Work through Sections 1.1 through 1.4 carefully, doing the exercises. If you find yourself struggling with basic simplification, factoring, or fraction manipulation, a targeted algebra review (focused only on those specific skills) is more efficient than reading an entire algebra textbook. The key algebra skills for calculus are: factoring, simplifying rational expressions, working with radicals and exponents, and solving equations.

Why does calculus use radians instead of degrees?

The derivative formulas for sine and cosine are clean only in radians: $\frac{d}{dx}\sin x = \cos x$. In degrees, a correction factor of $\frac{\pi}{180}$ appears everywhere, making every formula more complicated. Radians are the natural unit because the radian measure of an angle is defined as arc length divided by radius, which is a pure (dimensionless) number.

What is the difference between a function's domain and its natural domain?

The **domain** of a function is whatever set you declare as its allowable inputs. The **natural domain** (sometimes just called "the domain" when no restriction is stated) is the largest set of real numbers for which the function's formula gives a real-number output. For example, $f(x) = \sqrt{x}$ has natural domain $[0, \infty)$ because the square root of a negative number is not real. If you write $f(x) = \sqrt{x},\, x \ge 1$, you have restricted the domain to $[1, \infty)$.

Are inverse trig functions the same as the reciprocals of trig functions?

No, and confusing them is a very common error. The notation $\sin^{-1} x$(or$\arcsin x$) means the inverse function of sine, not $\frac{1}{\sin x}$. For example, $\sin^{-1}(1) = \frac{\pi}{2}$ because $\sin\frac{\pi}{2} = 1$. The reciprocal of $\sin x$ is written $(\sin x)^{-1} = \csc x$. This notational collision is unfortunate but deeply established, you simply have to know the context.