Numbers, Inequalities, and Absolute Values

Sets

Understanding sets is foundational in algebra. A set is a well-defined collection of objects, called elements. Sets can be described by listing elements, using ellipsis for large/infinite sets, or by set-builder notation:

• Listing: A = {1, 2, 3, 4}

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

• Membership: x \in A means x is an element of A.

• 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 is the set with no elements.

Example: Letters = {a, b, c, ..., z}

Real Numbers and Intervals

• 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 rational and irrational numbers.

• Irrational numbers: Numbers in \mathbb{R} \setminus \mathbb{Q}, e.g., \sqrt{2}, \pi.

Intervals: Subsets of \mathbb{R} representing all numbers between two endpoints.

Inequalities

Solving inequalities involves finding all real numbers that satisfy a given condition.

• Basic properties: If a < b and c > 0, then ac < bc. If c < 0, then ac > bc.

• Solution sets: Expressed in interval or set notation.

Example: Solve 3x + 1 > 2x:

• Subtract 2x: x + 1 > 0

• So x > -1

• Solution: (-1, \infty)

Absolute Value

The absolute value of a real number a is its distance from zero:

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

• Key properties:

  • |a| \geq 0

  • |a| = 0 \iff a = 0

  • |ab| = |a||b|

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

  • |a| = \sqrt{a^2}

Example: Solve |2x + 4| < 1:

• -1 < 2x + 4 < 1

• -5 < 2x < -3

• -\frac{5}{2} < x < -\frac{3}{2}

Functions

Definition and Basics

A function f: D \to Y assigns to each x \in D a unique f(x) \in Y. The set D is the domain, and the set of all images \{f(x) | x \in D\} is the range.

• Natural domain: All x for which f(x) is defined as a real number.

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

Graphing Functions

• The graph of f is the set \{(x, f(x)) | x \in D\}.

• Vertical Line Test: A curve is the graph of a function if every vertical line intersects it at most once.

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

Example: f(x) = \begin{cases} x^3 & x \leq 0 \\ 2x & 0 < x < 2 \\ x^2 & x \geq 2 \end{cases}

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 (symmetric about y-axis)

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

Examples: 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

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

• Algebraic: Built from algebraic operations (including roots)

Combining functions:

Composite and Inverse Functions

• Composite: (f \circ g)(x) = f(g(x)), domain: {x \in dom(g) | g(x) \in dom(f)}

• Inverse: f is one-to-one if f(x_1) = f(x_2) \implies x_1 = x_2. The inverse f^{-1} satisfies f(f^{-1}(y)) = y and f^{-1}(f(x)) = x.

• Horizontal Line Test: f is one-to-one if every horizontal line intersects its graph at most once.

Example: f(x) = 3x + 5 is one-to-one; f^{-1}(x) = \frac{x-5}{3}.

Angles and Trigonometric Functions

Radian Measure

• 1 radian is the angle subtended by an arc equal in length to the radius.

• Conversion:

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

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

• Arc length: s = r\alpha (α 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, 30°, 45°, 60°, 90° (see table below).

Trigonometric Identities

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

• Angle sum/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

• Product-to-sum: \sin \theta \cos \phi = \frac{1}{2} [\sin(\theta + \phi) + \sin(\theta - \phi)]

Inverse Trigonometric Functions

• arcsin x: The unique y in [-\frac{\pi}{2}, \frac{\pi}{2}] with \sin y = x, for x \in [-1, 1]

• arccos x: The unique y in [0, \pi] with \cos y = x, for x \in [-1, 1]

• arctan x: The unique y in (-\frac{\pi}{2}, \frac{\pi}{2}) with \tan y = x, for x \in \mathbb{R}

Trigonometric Equations

• General solution for sin θ = x: \theta = \arcsin x + 2k\pi or \theta = \pi - \arcsin x + 2k\pi, k \in \mathbb{Z}

• General solution for cos θ = x: \theta = \arccos x + 2k\pi or \theta = -\arccos x + 2k\pi

• General solution for tan θ = x: \theta = \arctan x + k\pi

Polar Coordinates and Graphs

Polar Coordinates

• Any point in the plane can be represented as (r, \theta), where r is the distance from the origin and θ is the angle from the positive x-axis.

• Conversion:

  • x = r \cos \theta

  • y = r \sin \theta

  • r = \sqrt{x^2 + y^2}

  • \tan \theta = \frac{y}{x}

Polar Equations and Graphs

• Circle: r = c

• Line through origin: \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:

• About x-axis: Replace θ with -θ

• About y-axis: Replace θ with π - θ

• About origin: Replace θ with θ + π

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

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

• Then a \cos x + b \sin x = R \cos(x - \theta)

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

Mathematical Induction

Principle of Mathematical Induction

• To prove a statement P(n) for all natural numbers n:

  1. Show P(1) is true (base case)

  2. Assume P(k) is true (inductive hypothesis), show P(k+1) is true (inductive step)

• If both steps hold, P(n) is true for all n ∈ ℕ.

Example: Prove 1 + 3 + 5 + ... + (2n-1) = n^2 for all n ∈ ℕ.

Sigma Notation and Binomial Theorem

Sigma Notation

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

• Properties:

  • \sum_{i=r}^n (a_i + b_i) = \sum_{i=r}^n a_i + \sum_{i=r}^n b_i

  • \sum_{i=r}^n k a_i = k \sum_{i=r}^n a_i

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)!}

Binomial Theorem

For any integer n ≥ 0,

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

Example: (x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4

Conic Sections

Quadratic Forms and Canonical Forms

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

• By completing the square and/or rotating axes, any non-degenerate conic can be written in canonical form:

  • 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

Key properties:

• Ellipse: Symmetric about both axes; if a = b, it's a circle.

• Hyperbola: Has asymptotes y = \pm \frac{b}{a} x.

• Parabola: Axis of symmetry determined by the linear term.

Example: x^2 + 2x + 2y^2 - 8y + 7 = 0 can be rewritten as \frac{(x+1)^2}{2} + (y-2)^2 = 1 (ellipse).

Appendix: Mathematical Reasoning

• Statement: A sentence that is either true or false.

• Implication: "If p, then q" (p \Rightarrow q); converse: q \Rightarrow p; contrapositive: \neg q \Rightarrow \neg p.

• Equivalence: p \Leftrightarrow q means both p \Rightarrow q and q \Rightarrow p are true.

• Proof methods: Direct, by contradiction, by induction.

Additional info: These notes are based on the structure and content of a first-semester college algebra manual, covering all foundational topics relevant to a standard college algebra course, including sets, numbers, inequalities, functions, trigonometry, induction, series, and conic sections. Examples and properties are expanded for clarity and exam preparation.