Precalculus Review: Essential Concepts and Functions

1.1 Algebra

Types of Numbers

Algebraic operations in precalculus are performed within the set of real numbers, which include several important subsets:

• Real Numbers ($\mathbb{R}$): All numbers on the number line, including positive and negative numbers, zero, fractions, and irrational numbers.

• Rational Numbers: Numbers that can be written as fractions, e.g., $\frac{1}{2}$, $-3$, $0.75$.

• Integers ($\mathbb{Z}$): Whole numbers and their negatives, e.g., $\ldots, -2, -1, 0, 1, 2, \ldots$.

• Natural Numbers: Positive integers, e.g., $1, 2, 3, \ldots$.

Infinity

Infinity ($\infty$) is a concept used to describe unbounded growth in mathematics. It is not a real number but is used to describe limits and behavior of functions as values grow larger or smaller without bound.

Intervals

Intervals are sets of real numbers between two endpoints. Common types include:

Interval Notation	Set Notation	Description
$(a, b)$	$a < x < b$	All numbers between $a$ and $b$, not including $a$ or $b$
$(a, b]$	$a < x \leq b$	All numbers between $a$ and $b$, including $b$
$[a, b)$	$a \leq x < b$	All numbers between $a$ and $b$, including $a$
$[a, b]$	$a \leq x \leq b$	All numbers between $a$ and $b$, including both endpoints

Absolute Value

The absolute value of $a$, denoted $|a|$, is defined as:

$|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$

Key properties include:

• $|ab| = |a||b|$

• $|a+b| \leq |a| + |b|$

Basic Algebra Formulas

Essential formulas for manipulating algebraic expressions:

• $(a+b)^2 = a^2 + 2ab + b^2$

• $(a-b)^2 = a^2 - 2ab + b^2$

• $(a+b)(a-b) = a^2 - b^2$

Solving Quadratic Equations

Quadratic equations of the form $ax^2 + bx + c = 0$ can be solved using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Fractions and Rational Expressions

Rational expressions are fractions involving polynomials. Simplification involves factoring and reducing common terms.

• Example: $\frac{2}{6} = \frac{1}{3}$

• Example: $\frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy}$

1.2 Functions and Graphs

Definition of a Function

A function is a rule that assigns to each input exactly one output. Functions are fundamental in mathematics and calculus.

Domain and Range

• Domain: The set of all possible input values ($x$) for which the function is defined.

• Range: The set of all possible output values ($f(x)$).

Substituting into Functions

To evaluate a function at a specific value, substitute the value into the function's formula.

Graphs of Functions

Graphs visually represent the relationship between input and output values. Common graphs include:

• $y = x$ (identity function)

• $y = x^2$ (parabola)

• $y = |x|$ (absolute value)

Transformations of Graphs

Transformation	Effect
$f(x) + a$	Shift up by $a$ units
$f(x) - a$	Shift down by $a$ units
$f(x+a)$	Shift left by $a$ units
$f(x-a)$	Shift right by $a$ units
$-f(x)$	Reflect across the $x$-axis
$f(-x)$	Reflect across the $y$-axis
$af(x)$	Vertical stretch by $a$
$f(ax)$	Horizontal compression by $a$

Piecewise Functions

Some functions are defined by different formulas on different parts of their domain. These are called piecewise functions.

Example: $f(x) = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$

1.3 Linear Functions

Slope and y-intercept

• Slope ($m$): Measures the steepness of a line. $m = \frac{y_2 - y_1}{x_2 - x_1}$

• y-intercept ($b$): The value where the line crosses the $y$-axis.

Equation of a Line

The general form is $y = mx + b$.

Parallel and Perpendicular Lines

• Parallel lines have equal slopes.

• Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$

1.4 Polynomials

Polynomial Functions

A polynomial function is defined as:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$

• Degree: The highest power of $x$.

• Roots: Values of $x$ where $f(x) = 0$.

Graphs of Polynomial Functions

Graphs of polynomials are smooth and continuous. The degree and leading coefficient determine the end behavior.

1.5 Power Functions

Definition

A power function has the form $f(x) = x^p$, where $p$ is a real number.

Negative and Fractional Powers

• $x^{-n} = \frac{1}{x^n}$

• $x^{1/n} = \sqrt[n]{x}$

Irrational Powers

Powers with irrational exponents are defined using limits and properties of exponents.

1.6 Trigonometric Functions

Radians and Degrees

• To convert degrees to radians: multiply by $\frac{\pi}{180}$

• To convert radians to degrees: multiply by $\frac{180}{\pi}$

The Six Trigonometric Functions

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\sin \theta}{\cos \theta}$

• $\csc \theta = \frac{1}{\sin \theta}$

• $\sec \theta = \frac{1}{\cos \theta}$

• $\cot \theta = \frac{1}{\tan \theta}$

Trigonometric Functions of Standard Angles

Angle	$\sin$	$\cos$
$0$	$0$	$1$
$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\frac{\pi}{2}$	$1$	$0$

Trigonometric Identities

• $\sin^2 x + \cos^2 x = 1$

• $1 + \tan^2 x = \sec^2 x$

• $1 + \cot^2 x = \csc^2 x$

1.7 Exponential Functions

Definition

An exponential function is $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

The Number $e$ and the Function $e^x$

• $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718$

• The function $f(x) = e^x$ is fundamental in calculus and applications.

Properties of Exponentials

• $a^x \cdot a^y = a^{x+y}$

• $\frac{a^x}{a^y} = a^{x-y}$

• $(a^x)^y = a^{xy}$

1.8 Logarithmic Functions

Definition

The logarithm with base $a$ is defined as $\log_a x = y$ if and only if $a^y = x$.

Common and Natural Logarithms

• Common logarithm: $\log_{10} x$

• Natural logarithm: $\ln x = \log_e x$

Properties of Logarithms

• $\log_a (xy) = \log_a x + \log_a y$

• $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

• $\log_a (x^r) = r \log_a x$

Graphs of Logarithmic Functions

The graph of $y = \log_a x$ is the inverse of the graph of $y = a^x$.

Additional info: This summary covers the main topics from the provided textbook pages, including algebraic fundamentals, functions and their graphs, linear and polynomial functions, power, trigonometric, exponential, and logarithmic functions, with definitions, properties, and key formulas. Tables have been recreated for intervals, transformations, and trigonometric values.