Numbers, Inequalities, and Absolute Values

1.1 Real Numbers

The real number system is foundational in algebra, encompassing several important subsets and notations.

• Natural Numbers (\(\mathbb{N}\)): \(\mathbb{N} = \{1, 2, 3, \ldots\}\)

• Integers (\(\mathbb{Z}\)): \(\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\)

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

• Irrational Numbers: Real numbers not expressible as a ratio of integers (e.g., \(\sqrt{2}, \pi\)).

• Real Numbers (\(\mathbb{R}\)): All rational and irrational numbers.

Ordering and notation:

• \(b > a\) means \(b\) is to the right of \(a\) on the number line.

• \(a \leq b\) means \(a < b\) or \(a = b\).

• Intervals are subsets of \(\mathbb{R}\):

1.2 Inequalities

Inequalities describe the relative size of numbers and are solved using algebraic manipulation.

• Solving Linear Inequalities: Manipulate as with equations, but reverse the inequality when multiplying/dividing by a negative.

• Solution Sets: Expressed in set or interval notation.

Example: Solve \(3x + 1 > 2x\).

• \(3x + 1 > 2x \implies x > -1\)

• Solution: \(( -1, \infty )\)

Example: Solve \(\frac{4}{x} + \frac{3}{x-2} \leq 1\).

• Combine into a single rational expression and analyze sign changes at critical points.

• Solution: \(( -\infty, 0 ) \cup [1, 2) \cup [8, \infty )\)

1.3 Absolute Values

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

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

• \(|a| \geq 0\) for all \(a\); \(|a| = 0\) if and only if \(a = 0\).

• \(|a| = \sqrt{a^2}\)

Key Properties:

• \(|ab| = |a||b|\)

• \(\left| \frac{a}{b} \right| = \frac{|a|}{|b|}\) if \(b \neq 0\)

• Triangle inequality: \(|a + b| \leq |a| + |b|\)

• \(|a| < b \iff -b < a < b\) 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

2.1 Functions: Some Basics

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

• Domain: Set of all inputs (\(D\)).

• Range: Set of all outputs (\(\{f(x) \mid x \in D\}\)).

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

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

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

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

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

2.1.1 Even and Odd Functions

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

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

Examples:

• Even: \(f(x) = x^2,\ f(x) = |x|\)

• Odd: \(f(x) = x^3,\ f(x) = x\)

2.2 Classification and Combination of Functions

2.2.1 Classification of Functions

• Polynomial: \(p(x) = a_n x^n + \ldots + a_0\), \(n\) a nonnegative integer.

• 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.

2.2.2 Sums, Differences, Products, and Quotients

2.2.3 Composite Functions

• \((f \circ g)(x) = f(g(x))\)

• Domain: \(\{x \in \text{dom}(g) \mid g(x) \in \text{dom}(f)\}\)

2.2.4 Shifting, Scaling, and Reflecting Graphs

• Translation: \(y = f(x + c)\) (horizontal), \(y = f(x) + c\) (vertical)

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

• Reflection: \(y = f(-x)\) (about y-axis), \(y = -f(x)\) (about x-axis)

2.3 Inverse Functions

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

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

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

• Graph of Inverse: Reflection of the graph of \(f\) across the line \(y = x\).

Example: \(f(x) = 3x + 5\) has inverse \(f^{-1}(x) = \frac{x - 5}{3}\).

Angles and Trigonometric Functions

3.1 Radian Measure

• Degrees: Circle divided into 360 parts.

• Radians: Based on arc length: \(\alpha = \frac{s}{r}\), where \(s\) is arc length, \(r\) is radius.

• Conversions:

  • Degrees to radians: Multiply by \(\frac{\pi}{180}\)

  • Radians to degrees: Multiply by \(\frac{180}{\pi}\)

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

• Area of Sector: \(A = \frac{1}{2} r^2 \alpha\)

3.2 Trigonometric Functions

• Defined for a point \((x, y)\) on a circle of radius \(r\):

• \(\sin \theta = \frac{y}{r}\), \(\cos \theta = \frac{x}{r}\), \(\tan \theta = \frac{y}{x}\)

• \(\csc \theta = \frac{r}{y}\), \(\sec \theta = \frac{r}{x}\), \(\cot \theta = \frac{x}{y}\)

Special Angles: Exact values for \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\).

Trigonometric Identities

• \(\sin^2 \theta + \cos^2 \theta = 1\)

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

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

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

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

3.3 Inverse Trigonometric Functions

• arcsin: Inverse of \(\sin\) on \(\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]\), domain \([-1, 1]\).

• arccos: Inverse of \(\cos\) on \([0, \pi]\), domain \([-1, 1]\).

• arctan: Inverse of \(\tan\) on \(\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\), domain \(\mathbb{R}\).

3.4 Trigonometric Equations

• General solutions use periodicity:

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

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

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

3.5 Polar Coordinates

• Point \((x, y)\) can be represented as \((r, \theta)\):

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

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

• Common polar graphs: circles, spirals, roses, cardioids.

3.6 Expressions of the Form \(a \cos x + b \sin x\)

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

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

Mathematical Induction

4.1 Introduction

Mathematical induction is a method of proof for statements indexed by the natural numbers.

• Base Case: Prove the statement for \(n = 1\).

• Inductive Step: Assume true for \(n = k\), prove for \(n = k + 1\).

If both steps are established, the statement holds for all \(n \in \mathbb{N}\).

4.2 Examples

• Sum of first \(n\) odd numbers: \(1 + 3 + 5 + \ldots + (2n - 1) = n^2\)

• Sum of squares: \(1^2 + 2^2 + \ldots + n^2 = \frac{1}{6} n(n + 1)(2n + 1)\)

Sigma Notation and Binomial Theorem

5.1 Sigma Notation

• \(\sum_{j=1}^n a_j = a_1 + a_2 + \ldots + a_n\)

• Index of summation is a dummy variable.

5.2 Factorials and Binomial Coefficients

• \(n! = n \times (n-1) \times \ldots \times 1\), \(0! = 1\)

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

• \(\binom{n}{k} = \binom{n}{n-k}\)

• Pascal's Rule: \(\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}\)

5.3 Binomial Theorem

For any nonnegative integer \(n\):

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

Coefficients correspond to entries in Pascal's Triangle.

Conic Sections

6.1 Quadratic Forms and Canonical Forms

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

• Non-degenerate conic sections can be reduced to canonical forms:

  • Parabola: \(y^2 = 4ax\)

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

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

6.2 Change of Axes

• Translation: Completing the square to shift the origin.

• Rotation: Remove cross term \(Bxy\) by rotating axes through angle \(\alpha\) where \(\cot 2\alpha = \frac{A - C}{B}\).

After suitable translation and rotation, any non-degenerate quadratic can be classified as a parabola, ellipse, or hyperbola.

Appendix: Basic Mathematical Notions

• Statements: Expressions that are either true or false.

• Definitions, Theorems, Axioms: Foundations of mathematical reasoning.

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

• Sets: Collections of objects, described by listing or set-builder notation.

• Subsets, Union, Intersection, Empty Set (\(\emptyset\)).

Additional info: The notes include detailed worked examples, tutorial problems, and references to further reading, supporting a comprehensive understanding of college algebra topics.