Numbers, Inequalities, and Absolute Values

Sets

Sets are fundamental collections of objects, called elements. Sets can be described by listing elements, using ellipsis for large sets, or by set-builder notation.

• Set notation: Curly brackets, e.g., A = {1, 2, 3, 4}.

• Set-builder notation: S = {x | A(x)}, where A(x) is a property of x.

• Subset: A \subset B if every element of A is in B.

• Intersection: A \cap B = {x | x \in A \text{ and } x \in B}.

• Union: A \cup B = {x | x \in A \text{ or } x \in B}.

• Empty set: \emptyset, the set with no elements.

Real Numbers

• Natural numbers: \mathbb{N} = {1, 2, 3, ...}

• Integers: \mathbb{Z} = {..., -2, -1, 0, 1, 2, ...}

• Rational numbers: \mathbb{Q} = \left\{ \frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0 \right\}

• Real numbers: \mathbb{R}, includes all rationals and irrationals.

• Irrational numbers: Real numbers not rational, e.g., \sqrt{2}, \pi.

Intervals

Intervals are subsets of \mathbb{R} representing all numbers between two endpoints.

Inequalities

• Solving inequalities: Find all x satisfying a given inequality, express solutions in interval or set notation.

• Example: Solve 3x + 1 > 2x:

  • 3x + 1 > 2x \implies x > -1, so solution is (-1, \infty).

Absolute Value

• Definition: |a| = \begin{cases} a & a \geq 0 \\ -a & a < 0 \end{cases}

• Key properties:

  • |a| \geq 0

  • |a|^2 = a^2

  • |ab| = |a||b|

  • |a + b| \leq |a| + |b| (Triangle Inequality)

  • |a| < b \iff -b < a < b for b \geq 0

• Example: Solve |2x + 4| < 1:

  • -1 < 2x + 4 < 1 \implies -5 < 2x < -3 \implies -\frac{5}{2} < x < -\frac{3}{2}

Functions

Definition and Basics

• Function: A rule assigning each x in domain D to a unique f(x) in codomain Y.

• Domain: Set of all inputs x for which f(x) is defined.

• Range: Set of all possible outputs f(x) as x varies over the domain.

• Example: f(x) = \frac{1}{\sqrt{x-1}} has domain (1, \infty).

Graphing Techniques

• Graph: Set of points (x, f(x)) in the plane.

• Vertical Line Test: A curve is the graph of a function if no vertical line intersects it more than once.

• Piecewise functions: Defined by different formulas on different parts of the domain.

• Increasing function: f(x_1) < f(x_2) whenever x_1 < x_2.

• Decreasing function: f(x_1) > f(x_2) whenever x_1 < x_2.

Transformations

• Shifts: y = f(x + c) (left/right), y = f(x) + c (up/down).

• Scaling: y = f(cx) (horizontal), y = cf(x) (vertical).

• Reflections: y = f(-x) (y-axis), y = -f(x) (x-axis).

Even and Odd Functions

• Even: f(-x) = f(x) for all x in domain. Graph is symmetric about y-axis.

• Odd: f(-x) = -f(x) for all x in domain. Graph is symmetric about the origin.

• Example: f(x) = x^2 is even; f(x) = x^3 is odd.

Classification and Combination of Functions

• Polynomial: p(x) = a_n x^n + ... + a_0, degree n.

• Rational: g(x) = \frac{p(x)}{q(x)}, where p, q are polynomials, q(x) \neq 0.

• Algebraic: Built from polynomials using roots and rational operations.

• Operations: (f+g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x), \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} (where g(x) \neq 0).

• Composite: (f \circ g)(x) = f(g(x)).

Inverse Functions

• One-to-one (injective): f(x_1) = f(x_2) \implies x_1 = x_2.

• Horizontal Line Test: Each horizontal line intersects the graph at most once.

• Inverse: If f is one-to-one, f^{-1}(y) = x such that f(x) = y.

• Graph: The graph of f^{-1} is the reflection of f across y = x.

Angles and Trigonometric Functions

Radian Measure

• Definition: The radian measure of an angle is the ratio of arc length s to radius r: \alpha = \frac{s}{r}.

• Conversion:

  • Degrees to radians: multiply by \frac{\pi}{180}

  • Radians to degrees: multiply by \frac{180}{\pi}

• Arc length: s = r\alpha (with \alpha in radians)

• Area of sector: A = \frac{1}{2} r^2 \alpha

Trigonometric Functions

• Definitions (unit circle):

  • \sin \theta = \frac{y}{r}

  • \cos \theta = \frac{x}{r}

  • \tan \theta = \frac{y}{x}

  • \csc \theta = \frac{r}{y}

  • \sec \theta = \frac{r}{x}

  • \cot \theta = \frac{x}{y}

• Special angles: Know exact values for 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}.

• Periodicity:

  • \sin, \cos, \sec, \csc have period 2\pi

  • \tan, \cot have period \pi

Trigonometric Identities

• Reciprocal: \sec \theta = \frac{1}{\cos \theta}, \csc \theta = \frac{1}{\sin \theta}

• Quotient: \tan \theta = \frac{\sin \theta}{\cos \theta}, \cot \theta = \frac{\cos \theta}{\sin \theta}

• Pythagorean: \sin^2 \theta + \cos^2 \theta = 1

• Sum and difference:

  • \sin(\theta \pm \phi) = \sin \theta \cos \phi \pm \cos \theta \sin \phi

  • \cos(\theta \pm \phi) = \cos \theta \cos \phi \mp \sin \theta \sin \phi

• Double angle:

  • \sin(2\theta) = 2 \sin \theta \cos \theta

  • \cos(2\theta) = \cos^2 \theta - \sin^2 \theta

• Power-reduction:

  • \sin^2 \theta = \frac{1 - \cos 2\theta}{2}

  • \cos^2 \theta = \frac{1 + \cos 2\theta}{2}

Inverse Trigonometric Functions

• arcsin: \arcsin x is the unique y \in [-\frac{\pi}{2}, \frac{\pi}{2}] with \sin y = x.

• arccos: \arccos x is the unique y \in [0, \pi] with \cos y = x.

• arctan: \arctan x is the unique y \in (-\frac{\pi}{2}, \frac{\pi}{2}) with \tan y = x.

• Domains and ranges: Know the principal values for each inverse function.

Trigonometric Equations

• General solutions:

  • \sin \theta = x \implies \theta = \arcsin x + 2k\pi or \theta = \pi - \arcsin x + 2k\pi

  • \cos \theta = x \implies \theta = \arccos x + 2k\pi or \theta = -\arccos x + 2k\pi

  • \tan \theta = x \implies \theta = \arctan x + k\pi

Polar Coordinates and Graphs

• Conversion:

  • x = r \cos \theta, y = r \sin \theta

  • r = \sqrt{x^2 + y^2}, \tan \theta = \frac{y}{x}

• Common polar graphs:

  • Circle: r = c

  • Line: \theta = k

  • Spiral: r = k\theta

  • Rose: r = a \cos n\theta or r = a \sin n\theta

  • Cardioid: r = a + b \sin \theta or r = a + b \cos \theta

• Symmetry tests:

  • x-axis: Replace \theta with -\theta

  • y-axis: Replace \theta with \pi - \theta

  • Origin: Replace \theta with \pi + \theta

Expressing a \cos x + b \sin x as R \cos(x - \theta)

• Let a = R \cos \theta, b = R \sin \theta, then R = \sqrt{a^2 + b^2}, \tan \theta = \frac{b}{a}.

• Example: \cos x + \sqrt{3} \sin x = 2 \cos(x - \frac{\pi}{3})

Mathematical Induction, Sigma Notation, and Binomial Theorem

Mathematical Induction

• Principle: To prove a statement for all n \in \mathbb{N}:

  1. Show it is true for n = 1 (base case).

  2. Assume true for n = k (inductive hypothesis), prove true for n = k + 1.

• Example: Prove 1 + 2 + ... + n = \frac{n(n+1)}{2} for all n \in \mathbb{N}.

Sigma Notation

• Definition: \sum_{j=1}^n a_j = a_1 + a_2 + ... + a_n

• Properties:

  • Linearity: \sum (a_j + b_j) = \sum a_j + \sum b_j

  • Constants: \sum k = nk

• Common sums:

  • \sum_{r=1}^n r = \frac{n(n+1)}{2}

  • \sum_{r=1}^n r^2 = \frac{n(n+1)(2n+1)}{6}

  • \sum_{r=1}^n r^3 = \frac{n^2(n+1)^2}{4}

Factorials and Binomial Coefficients

• Factorial: n! = n \times (n-1) \times ... \times 1, 0! = 1

• Binomial coefficient: \binom{n}{k} = \frac{n!}{k!(n-k)!}

• Properties:

  • \binom{n}{0} = \binom{n}{n} = 1

  • \binom{n}{k} = \binom{n}{n-k}

  • \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}

Binomial Theorem

• Statement: (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k

• Pascal's Triangle: The coefficients \binom{n}{k} form Pascal's triangle.

• Applications: Expanding powers, finding coefficients, and combinatorics.

Conic Sections

Quadratic Forms and Canonical Forms

• General quadratic: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

• Canonical forms:

  • Parabola: y^2 = 4ax

  • Ellipse: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

  • Hyperbola: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Classification

• Parabola: One squared term, e.g., y^2 = 4ax

• Ellipse: Both squared terms, same sign, e.g., \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

• Circle: Special ellipse, a = b

• Hyperbola: Both squared terms, opposite signs, e.g., \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Change of Axes

• Translation: Completing the square to shift the origin.

• Rotation: Removing the xy term by rotating axes through angle \alpha where \cot 2\alpha = \frac{A - C}{B}.

Summary Table: Conic Sections

Additional info: These notes are based on the lecture manual for a first-semester precalculus algebra course, covering foundational topics in algebra, functions, trigonometry, and conic sections. For more advanced topics (e.g., sequences, series, matrices, limits), refer to subsequent chapters or modules.