College Algebra: Core Concepts, Functions, Trigonometry, Induction, and Conic Sections

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/infinite sets, or by set-builder notation:

• Notation: S = {x | A(x)} means S is the set of all x for which A(x) is true.

• Subset: A ⊂ B if every element of A is also in B.

• Intersection: A ∩ B = {x | x ∈ A and x ∈ B}.

• Union: A ∪ B = {x | x ∈ A or x ∈ B}.

• Empty set: ∅ denotes the set with no elements.

Real Numbers and Intervals

• Natural numbers (N): {1, 2, 3, ...}

• Integers (Z): {..., -2, -1, 0, 1, 2, ...}

• Rational numbers (Q): All numbers of the form $\frac{p}{q}$, with $p, q \in \mathbb{Z}$, $q \neq 0$.

• Real numbers (R): All points on the number line, including rationals and irrationals.

• Irrational numbers: Real numbers not expressible as $\frac{p}{q}$.

Intervals are subsets of the real line, denoted as:

Notation	Set Description	Type
(a, b)	{x ∈ R | a < x < b}	Open
[a, b]	{x ∈ R | a ≤ x ≤ b}	Closed
[a, b)	{x ∈ R | a ≤ x < b}	Half-open
(a, b]	{x ∈ R | a < x ≤ b}	Half-open
(a, ∞)	{x ∈ R | x > a}	Open
[a, ∞)	{x ∈ R | x ≥ a}	Closed

Inequalities

• Solving inequalities: Find all real numbers x satisfying a given inequality.

• Notation: $a < b$, $a \leq b$, $a > b$, $a \geq b$.

• Solution sets: Expressed in interval or set notation.

Example: Solve $3x + 1 > 2x$:

• $3x + 1 > 2x \implies x > -1$

• Solution: $x \in (-1, \infty)$

Absolute Value

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

• Key properties:

  • $|a| \geq 0$ for all $a$

  • $|a| = 0$ iff $a = 0$

  • $|ab| = |a||b|$

  • $|a + b| \leq |a| + |b|$ (triangle inequality)

  • $|a| = \sqrt{a^2}$

• Solving absolute value equations/inequalities:

  • $|x| < b \iff -b < x < b$

  • $|x| > b \iff x < -b$ or $x > b$

Functions

Definition and Basics

• A function $f$ from set $D$ to $Y$ assigns each $x \in D$ a unique $f(x) \in Y$.

• Domain: Set of all possible inputs (x-values).

• Range: Set of all possible outputs (f(x)-values).

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

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

Graphing Functions

• The graph of $f$ is the set of points $(x, f(x))$ for $x$ in the domain.

• 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 intervals.

Transformations

• Shifts: $y = f(x + c)$ (horizontal), $y = f(x) + c$ (vertical)

• 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).

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

Classification and Combination of Functions

• Polynomial: $p(x) = a_n x^n + \dots + a_0$

• Rational: $g(x) = \frac{p(x)}{q(x)}$, $q(x) \neq 0$

• 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 $g(x) \neq 0$)

• Composition: $(f \circ g)(x) = f(g(x))$

Inverse Functions

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

• Inverse: $f^{-1}(y) = x$ such that $f(x) = y$

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

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

Angles and Trigonometric Functions

Radian Measure

• Definition: The radian measure of an angle is $\alpha = \frac{s}{r}$, where $s$ is arc length and $r$ is radius.

• Conversions:

  • 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

• 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:

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

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

• Product-to-sum and sum-to-product formulas (see notes for details).

Inverse Trigonometric Functions

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

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

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

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

Trigonometric Equations

• General solutions use periodicity and principal values:

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

• Definition: $(r, \theta)$ where $r \geq 0$ is the distance from the origin, $\theta$ 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: Equations in $r$ and $\theta$ can describe circles, lines, spirals, roses, etc.

• Symmetry tests:

  • About x-axis: replace $\theta$ with $-\theta$

  • About y-axis: replace $\theta$ with $\pi - \theta$

  • About origin: replace $\theta$ with $\pi + \theta$

Expressions of the Form $a \cos x + b \sin x$

• Can be rewritten as $R \cos(x - \theta)$ where $R = \sqrt{a^2 + b^2}$ and $\tan \theta = \frac{b}{a}$.

• 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 $n \in \mathbb{N}$:

• Base case: Prove $P(1)$ is true.

• Inductive step: Assume $P(k)$ is true, prove $P(k+1)$ is true.

• If both steps hold, $P(n)$ is true for all $n \in \mathbb{N}$.

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

Sigma Notation and Binomial Theorem

Sigma Notation

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

• 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: $n! = n \cdot (n-1) \cdots 2 \cdot 1$, $0! = 1$

• 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

• For any integer $n \geq 0$:

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

• Coefficients correspond to entries in Pascal's Triangle.

• Example: $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$

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 quadratic can be reduced to one of:

Curve	Canonical Equation
Parabola	$y^2 = 4ax$
Ellipse	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
Hyperbola	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

• Ellipse: Symmetric about both axes, bounded.

• Hyperbola: Two branches, asymptotes $y = \pm \frac{b}{a} x$.

• Parabola: One quadratic and one linear term.

Change of Axes

• Translation: Shifts the origin to simplify the equation.

• Rotation: Removes the $xy$ term; angle $\alpha$ found by $\cot 2\alpha = \frac{A - C}{B}$.

• After transformation, the equation can be classified as parabola, ellipse, or hyperbola.

Appendix: Basic Mathematical Notions

• Statement: An expression that is either true or false.

• Theorem: A true statement proved from axioms and definitions.

• Axiom: A statement assumed true without proof.

• Implication: "If p then q" ($p \implies q$); converse and contrapositive.

• Proof by induction: See above.

Additional info: This summary covers the foundational topics in college algebra, including sets, numbers, inequalities, functions, trigonometry, induction, and conic sections, as outlined in the provided lecture manual. For more advanced or detailed examples, refer to the full lecture notes or prescribed textbook.