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 (\(\mathbb{N}\)): \(\{1, 2, 3, \ldots\}\)

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

• Rational Numbers (\(\mathbb{Q}\)): Numbers expressible as \(\frac{p}{q}\), where \(p, q \in \mathbb{Z}, q \neq 0\).

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

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

Ordering and notation:

• \(a < b\): \(a\) is less than \(b\).

• \(a \leq b\): \(a\) is less than or equal to \(b\).

• Intervals: Open: \((a, b) = \{x \in \mathbb{R} \mid a < x < b\}\) Closed: \([a, b] = \{x \in \mathbb{R} \mid a \leq x \leq b\}\)

Example: The intersection \((−2, 10) \cap (−10, 2) = (−2, 2)\).

1.2 Inequalities

Solving inequalities involves finding all real numbers that satisfy given conditions.

• Key Steps:

  1. Isolate the variable.

  2. Consider the domain (e.g., denominators not zero).

  3. Express solutions in interval and set notation.

Example: Solve \(3x + 1 > 2x\): \(3x + 1 > 2x \implies x > -1\), so solution is \((−1, \infty)\).

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 \(a = 0\).

• \(|a| = \sqrt{a^2}\)

Properties:

• \(|ab| = |a||b|\)

• \(|a + b| \leq |a| + |b|\) (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

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 possible inputs (\(x\)).

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

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

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

2.1.1 Even and Odd Functions

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

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

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

2.2 Classification and Combination of Functions

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

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

• Algebraic Function: Built from algebraic operations (addition, subtraction, multiplication, division, roots).

Operations:

2.2.3 Composite Functions

The composite \(f \circ g\) is defined by \((f \circ g)(x) = f(g(x))\). The domain is all \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f\).

2.2.4 Transformations of Graphs

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

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

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

2.3 Inverse Functions

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

• Inverse Function: 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.

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

Angles and Trigonometric Functions

3.1 Radian Measure

• 1 radian: The angle subtended at the center of a circle 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: \(s = r\alpha\), where \(\alpha\) is in radians.

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

3.2 Trigonometric Functions

• \(\sin \theta = \frac{y}{r}\)

• \(\cos \theta = \frac{x}{r}\)

• \(\tan \theta = \frac{y}{x}\)

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

Special Angles Table:

3.2.4 Trigonometric Identities

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

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

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

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

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

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

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

3.3 Inverse Trigonometric Functions

• arcsin: Domain \([-1, 1]\), Range \([ -\frac{\pi}{2}, \frac{\pi}{2} ]\)

• arccos: Domain \([-1, 1]\), Range \([0, \pi]\)

• arctan: Domain \(\mathbb{R}\), Range \(( -\frac{\pi}{2}, \frac{\pi}{2} )\)

Example: \(\arcsin \frac{1}{2} = \frac{\pi}{6}\), \(\arccos \frac{1}{2} = \frac{\pi}{3}\), \(\arctan 1 = \frac{\pi}{4}\).

3.4 Trigonometric Equations

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

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

• General solution for \(\tan \theta = x\): \(\theta = \arctan x + k\pi\).

3.5 Polar Coordinates

• Cartesian: \((x, y)\)

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

• \(r = \sqrt{x^2 + y^2}\), \(\theta = \arctan \frac{y}{x}\) (quadrant considerations apply)

Example: The equation \(r = 2\) in polar coordinates is \(x^2 + y^2 = 4\) in Cartesian coordinates.

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

Any expression \(a \cos x + b \sin x\) 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

4.2 Principle of Mathematical Induction

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

  1. Prove the base case (usually \(n = 1\)).

  2. Assume true for \(n = k\) (inductive hypothesis).

  3. Prove true for \(n = k + 1\).

Example: Prove \(1 + 3 + 5 + \ldots + (2n - 1) = n^2\) for all \(n \in \mathbb{N}\).

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.

Common Sums:

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

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

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

5.2 Factorials and Binomial Coefficients

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

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

5.3 Binomial Theorem

For any nonnegative integer \(n\):

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

Example: Expand \((x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\).

Conic Sections

6.1 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:

• Ellipse: Both quadratic terms have the same sign.

• Hyperbola: Quadratic terms have opposite signs.

• Parabola: Only one variable is squared.

6.2 Change of Axes

• Translation: Completing the square to shift the origin.

• Rotation: Used to eliminate the \(xy\) term. The rotation angle \(\alpha\) satisfies \(\cot 2\alpha = \frac{A - C}{B}\).

Appendix: Basic Mathematical Notions

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

• Set: A well-defined collection of objects.

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

• Union: \(A \cup B\) is the set of elements in \(A\) or \(B\).

• Intersection: \(A \cap B\) is the set of elements in both \(A\) and \(B\).

• Empty Set: The set with no elements, denoted \(\emptyset\).

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