Pre-Calculus

Comprehensive pre-calculus guide covering advanced functions, trigonometry, sequences, limits, and mathematical analysis fundamentals.

pre-calculus functions trigonometry limits sequences logarithms exponentials conic-sections

Prerequisites

Required:

Algebra - All topics including functions, graphing, polynomials, factoring
Geometry - For trigonometry and coordinate geometry

Helpful:

Strong comfort with algebraic manipulation
Understanding of basic geometric relationships

Overview

Pre-Calculus bridges the gap between Algebra and Calculus, focusing on advanced functions and mathematical analysis. It develops the sophisticated understanding of functions, their behavior, and their relationships that forms the foundation for calculus and higher mathematics.

This comprehensive guide covers polynomial and rational functions, exponential and logarithmic functions, trigonometry, sequences and series, and introduces the concept of limits. Master these concepts to succeed in calculus and apply mathematics to science, engineering, and advanced problem-solving.

Advanced Functions

Function Composition and Operations

Function Operations

Given f(x) and g(x):

Addition: (f + g)(x) = f(x) + g(x)
Subtraction: (f - g)(x) = f(x) - g(x)
Multiplication: (f · g)(x) = f(x) · g(x)
Division: (f/g)(x) = f(x)/g(x), where g(x) ≠ 0

Example: f(x) = x² + 1, g(x) = 2x - 3
(f + g)(x) = x² + 1 + 2x - 3 = x² + 2x - 2
(f · g)(x) = (x² + 1)(2x - 3) = 2x³ - 3x² + 2x - 3

Function Composition

(f ∘ g)(x) = f(g(x))
Read as "f composed with g" or "f of g of x"

Example: f(x) = x² + 1, g(x) = 2x - 3
(f ∘ g)(x) = f(2x - 3) = (2x - 3)² + 1 = 4x² - 12x + 9 + 1 = 4x² - 12x + 10

Order matters:
(g ∘ f)(x) = g(x² + 1) = 2(x² + 1) - 3 = 2x² + 2 - 3 = 2x² - 1

Inverse Functions

f⁻¹ is the inverse of f if:
f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

To find inverse:
1. Replace f(x) with y
2. Swap x and y  
3. Solve for y
4. Replace y with f⁻¹(x)

Example: f(x) = 2x + 3
Step 1: y = 2x + 3
Step 2: x = 2y + 3
Step 3: x - 3 = 2y, so y = (x - 3)/2
Step 4: f⁻¹(x) = (x - 3)/2

Verification: f(f⁻¹(x)) = f((x-3)/2) = 2((x-3)/2) + 3 = x - 3 + 3 = x ✓

Properties of Inverse Functions

• Graphs are reflections across y = x
• Domain of f = Range of f⁻¹
• Range of f = Domain of f⁻¹
• Only one-to-one functions have inverses
• Horizontal Line Test: each horizontal line intersects graph at most once

Polynomial Functions

General Form and Behavior

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
where aₙ ≠ 0, n is the degree

Leading coefficient: aₙ
Constant term: a₀

End behavior depends on degree and leading coefficient:
• Even degree, positive leading coefficient: both ends up
• Even degree, negative leading coefficient: both ends down  
• Odd degree, positive leading coefficient: left down, right up
• Odd degree, negative leading coefficient: left up, right down

Zeros and Factoring

Fundamental Theorem of Algebra:
A polynomial of degree n has exactly n complex zeros (counting multiplicity)

Factor Theorem: (x - a) is a factor of P(x) if and only if P(a) = 0

Rational Root Theorem: If p/q is a rational zero of P(x), then:
• p divides the constant term
• q divides the leading coefficient

Example: P(x) = 2x³ - 5x² + x + 2
Possible rational zeros: ±1, ±2, ±1/2
Test P(2) = 16 - 20 + 2 + 2 = 0, so (x - 2) is a factor

Multiplicity of Zeros

If (x - a)ᵏ is a factor of P(x), then a has multiplicity k:

• Odd multiplicity: graph crosses x-axis at zero
• Even multiplicity: graph touches x-axis but doesn't cross

Example: P(x) = (x - 1)²(x + 2)³
• Zero at x = 1 with multiplicity 2 (touches x-axis)
• Zero at x = -2 with multiplicity 3 (crosses x-axis)

Rational Functions

General Form

R(x) = P(x)/Q(x)
where P(x) and Q(x) are polynomials, Q(x) ≠ 0

Domain: All real numbers except zeros of Q(x)

Example: R(x) = (x² - 1)/(x² - 4)
Domain: x ≠ ±2 (since x² - 4 = 0 when x = ±2)

Vertical Asymptotes

Occur at zeros of denominator that are not zeros of numerator

Example: R(x) = (x + 1)/((x - 2)(x + 3))
Vertical asymptotes at x = 2 and x = -3

Behavior near vertical asymptote:
• Function approaches +∞ or -∞
• Check signs on either side to determine direction

Horizontal Asymptotes

Depend on degrees of numerator (m) and denominator (n):

Case 1: m < n → Horizontal asymptote at y = 0
Case 2: m = n → Horizontal asymptote at y = aₘ/bₙ (ratio of leading coefficients)  
Case 3: m > n → No horizontal asymptote (but may have oblique asymptote)

Example 1: R(x) = (2x + 1)/(x² + 3x - 1)
m = 1 < n = 2, so horizontal asymptote at y = 0

Example 2: R(x) = (3x² + 2x - 1)/(2x² - x + 5)  
m = n = 2, so horizontal asymptote at y = 3/2

Holes vs. Vertical Asymptotes

If (x - a) is a factor of both numerator and denominator:
• Cancel the common factor
• Result is a hole at x = a, not a vertical asymptote

Example: R(x) = (x² - 1)/(x - 1) = (x + 1)(x - 1)/(x - 1) = x + 1 (x ≠ 1)
Hole at (1, 2), not a vertical asymptote

Exponential and Logarithmic Functions

Exponential Functions

Definition and Properties

f(x) = aˣ where a > 0, a ≠ 1

Properties:
• Domain: (-∞, ∞)
• Range: (0, ∞)
• y-intercept: (0, 1) since a⁰ = 1
• Horizontal asymptote: y = 0

If a > 1: increasing function (growth)
If 0 < a < 1: decreasing function (decay)

Natural Exponential Function

f(x) = eˣ where e ≈ 2.71828...

e is defined as: lim(n→∞)(1 + 1/n)ⁿ

Properties:
• Base of natural logarithm
• Rate of change equals the function value
• Appears naturally in growth/decay problems
• Most important exponential function in calculus

Exponential Growth and Decay

General model: A(t) = A₀e^(kt)
• A₀ = initial amount
• k = growth rate (k > 0) or decay rate (k < 0)
• t = time

Applications:
• Population growth: P(t) = P₀e^(rt)
• Radioactive decay: N(t) = N₀e^(-λt)
• Compound interest: A = Pe^(rt)
• Newton's law of cooling: T(t) = Tₐ + (T₀ - Tₐ)e^(-kt)

Logarithmic Functions

Definition

y = logₐ(x) if and only if aʸ = x
where a > 0, a ≠ 1, x > 0

Common logarithms: log(x) = log₁₀(x)
Natural logarithms: ln(x) = logₑ(x)

Properties of Logarithms

Product rule: logₐ(xy) = logₐ(x) + logₐ(y)
Quotient rule: logₐ(x/y) = logₐ(x) - logₐ(y)
Power rule: logₐ(xⁿ) = n·logₐ(x)

Special values:
logₐ(1) = 0 (since a⁰ = 1)
logₐ(a) = 1 (since a¹ = a)
logₐ(aˣ) = x (inverse property)
a^(logₐ(x)) = x (inverse property)

Change of Base Formula

logₐ(x) = logᵦ(x)/logᵦ(a) = ln(x)/ln(a) = log(x)/log(a)

Example: log₃(17) = ln(17)/ln(3) ≈ 2.833/1.099 ≈ 2.58

Solving Exponential and Logarithmic Equations

Exponential equations:
aˣ = b → x = logₐ(b)

Example: 3ˣ = 81
3ˣ = 3⁴
x = 4

Or: 2ˣ = 10
x = log₂(10) = ln(10)/ln(2) ≈ 3.32

Logarithmic equations:
logₐ(x) = b → x = aᵇ

Example: log₂(x) = 5
x = 2⁵ = 32

Complex example: log(x + 1) + log(x - 1) = log(8)
log((x + 1)(x - 1)) = log(8)
x² - 1 = 8
x² = 9
x = 3 (x = -3 invalid since x > 1 required)

Trigonometry

Angle Measurement

Degrees vs. Radians

1 complete revolution = 360° = 2π radians

Conversion formulas:
Degrees to radians: multiply by π/180
Radians to degrees: multiply by 180/π

Common angles:
30° = π/6 radians    60° = π/3 radians    90° = π/2 radians
45° = π/4 radians    120° = 2π/3 radians   180° = π radians

Arc Length and Sector Area

For central angle θ (in radians) and radius r:
Arc length: s = rθ
Sector area: A = ½r²θ

Example: Circle with radius 8, central angle 3π/4
Arc length = 8 · (3π/4) = 6π
Sector area = ½ · 64 · (3π/4) = 24π

Trigonometric Functions

Right Triangle Definitions

For acute angle θ in right triangle:

sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent

csc(θ) = 1/sin(θ) = hypotenuse/opposite
sec(θ) = 1/cos(θ) = hypotenuse/adjacent
cot(θ) = 1/tan(θ) = adjacent/opposite

Unit Circle Definitions

For angle θ in standard position (vertex at origin, initial side on positive x-axis):
Point on unit circle: (cos(θ), sin(θ))

sin(θ) = y-coordinate
cos(θ) = x-coordinate  
tan(θ) = sin(θ)/cos(θ) = y/x

This extends definitions to all real numbers, not just acute angles

Special Angle Values

θ        sin(θ)    cos(θ)    tan(θ)
0°/0      0         1         0
30°/π/6   1/2       √3/2      √3/3
45°/π/4   √2/2      √2/2      1
60°/π/3   √3/2      1/2       √3
90°/π/2   1         0         undefined

These form the foundation for all other angle calculations

Signs of Trig Functions by Quadrant

Quadrant I (0° to 90°): All positive
Quadrant II (90° to 180°): sin positive, cos and tan negative
Quadrant III (180° to 270°): tan positive, sin and cos negative  
Quadrant IV (270° to 360°): cos positive, sin and tan negative

Memory device: "All Students Take Calculus"

Trigonometric Identities

Fundamental Identities

Pythagorean identities:
sin²(θ) + cos²(θ) = 1
1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)

Reciprocal identities:
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)

Quotient identities:
tan(θ) = sin(θ)/cos(θ)
cot(θ) = cos(θ)/sin(θ)

Sum and Difference Formulas

sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
tan(A ± B) = (tan(A) ± tan(B))/(1 ∓ tan(A)tan(B))

Example: sin(75°) = sin(45° + 30°)
= sin(45°)cos(30°) + cos(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4

Double Angle Formulas

sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
tan(2θ) = 2tan(θ)/(1 - tan²(θ))

Example: If sin(θ) = 3/5 and θ in Quadrant I:
cos(θ) = 4/5 (from Pythagorean theorem)
sin(2θ) = 2(3/5)(4/5) = 24/25
cos(2θ) = (4/5)² - (3/5)² = 16/25 - 9/25 = 7/25

Half Angle Formulas

sin(θ/2) = ±√((1 - cos(θ))/2)
cos(θ/2) = ±√((1 + cos(θ))/2)
tan(θ/2) = ±√((1 - cos(θ))/(1 + cos(θ))) = sin(θ)/(1 + cos(θ))

Sign depends on quadrant of θ/2

Graphs of Trigonometric Functions

Sine and Cosine Functions

f(x) = A sin(Bx + C) + D
g(x) = A cos(Bx + C) + D

• A = amplitude (vertical stretch)
• B affects period: Period = 2π/|B|
• C affects phase shift: shift = -C/B  
• D = vertical shift

Example: f(x) = 3sin(2x - π) + 1
• Amplitude = 3
• Period = 2π/2 = π
• Phase shift = -(-π)/2 = π/2 right
• Vertical shift = 1 up

Tangent Function

f(x) = A tan(Bx + C) + D

• Period = π/|B|
• Vertical asymptotes where cos(Bx + C) = 0
• No amplitude (function has infinite range)

Example: f(x) = 2tan(x/2)
• Period = π/(1/2) = 2π
• Vertical asymptotes at x = π + 2πn

Inverse Trigonometric Functions

y = arcsin(x) or sin⁻¹(x): Domain [-1,1], Range [-π/2, π/2]
y = arccos(x) or cos⁻¹(x): Domain [-1,1], Range [0, π]
y = arctan(x) or tan⁻¹(x): Domain (-∞,∞), Range (-π/2, π/2)

Example: arcsin(1/2) = π/6 = 30°
arccos(√2/2) = π/4 = 45°
arctan(√3) = π/3 = 60°

Note: These give principal values (specific ranges)

Sequences and Series

Sequences

Arithmetic Sequences

Definition: Constant difference between consecutive terms

General term: aₙ = a₁ + (n-1)d
where a₁ = first term, d = common difference

Example: 3, 7, 11, 15, 19, ...
a₁ = 3, d = 4
aₙ = 3 + (n-1)4 = 4n - 1
a₁₀ = 4(10) - 1 = 39

Geometric Sequences

Definition: Constant ratio between consecutive terms

General term: aₙ = a₁ · r^(n-1)
where a₁ = first term, r = common ratio

Example: 2, 6, 18, 54, 162, ...
a₁ = 2, r = 3
aₙ = 2 · 3^(n-1)
a₆ = 2 · 3⁵ = 2 · 243 = 486

Series

Arithmetic Series

Sum of arithmetic sequence:
Sₙ = n/2(a₁ + aₙ) = n/2(2a₁ + (n-1)d)

Example: Sum of first 20 terms of 3, 7, 11, 15, ...
S₂₀ = 20/2(2·3 + (20-1)·4) = 10(6 + 76) = 820

Or: a₂₀ = 4(20) - 1 = 79
S₂₀ = 20/2(3 + 79) = 10(82) = 820

Geometric Series

Sum of finite geometric series:
Sₙ = a₁(1 - rⁿ)/(1 - r) where r ≠ 1

Sum of infinite geometric series (|r| < 1):
S∞ = a₁/(1 - r)

Example 1: Sum of 2 + 6 + 18 + 54 + 162
S₅ = 2(1 - 3⁵)/(1 - 3) = 2(-242)/(-2) = 242

Example 2: Sum of 1 + 1/2 + 1/4 + 1/8 + ...
S∞ = 1/(1 - 1/2) = 1/(1/2) = 2

Applications of Series

Compound Interest: A = P(1 + r)ⁿ forms geometric sequence
Depreciation: Value forms geometric sequence with r < 1
Population Growth: Often modeled as geometric sequence
Fractals: Self-similar patterns use geometric series

Conic Sections

Circle

Standard form: (x - h)² + (y - k)² = r²
Center: (h, k), Radius: r

General form: x² + y² + Dx + Ey + F = 0
Complete the square to convert to standard form

Example: x² + y² - 6x + 4y - 3 = 0
(x² - 6x + 9) + (y² + 4y + 4) = 3 + 9 + 4
(x - 3)² + (y + 2)² = 16
Center: (3, -2), Radius: 4

Parabola

Vertex form: (x - h)² = 4p(y - k) [opens up/down]
            (y - k)² = 4p(x - h) [opens left/right]

Vertex: (h, k)
Focus: (h, k + p) or (h + p, k)  
Directrix: y = k - p or x = h - p

Example: x² = 8y
4p = 8, so p = 2
Vertex: (0, 0), Focus: (0, 2), Directrix: y = -2

Ellipse

Standard form: (x - h)²/a² + (y - k)²/b² = 1
Center: (h, k)

If a > b: horizontal major axis
• Vertices: (h ± a, k)
• Co-vertices: (h, k ± b)  
• Foci: (h ± c, k) where c² = a² - b²

If b > a: vertical major axis
• Vertices: (h, k ± b)
• Co-vertices: (h ± a, k)
• Foci: (h, k ± c) where c² = b² - a²

Eccentricity: e = c/a (for major axis), measures how "stretched"

Hyperbola

Standard form: (x - h)²/a² - (y - k)²/b² = 1 [horizontal]
              (y - k)²/a² - (x - h)²/b² = 1 [vertical]

Center: (h, k)
Vertices: (h ± a, k) [horizontal] or (h, k ± a) [vertical]
Foci: (h ± c, k) or (h, k ± c) where c² = a² + b²
Asymptotes: y - k = ±(b/a)(x - h)

Example: x²/9 - y²/16 = 1
a² = 9, b² = 16, so a = 3, b = 4, c = 5
Vertices: (±3, 0), Foci: (±5, 0)
Asymptotes: y = ±(4/3)x

Limits and Continuity

Limit Concept

lim[x→c] f(x) = L means:
"As x approaches c, f(x) approaches L"

This may or may not equal f(c)

Types of limits:
• Two-sided: lim[x→c] f(x)
• Left-sided: lim[x→c⁻] f(x) 
• Right-sided: lim[x→c⁺] f(x)
• At infinity: lim[x→∞] f(x)

Evaluating Limits

Method 1: Direct substitution (if function is continuous)
lim[x→2] (x² + 3x - 1) = 4 + 6 - 1 = 9

Method 2: Factoring (for rational functions)
lim[x→3] (x² - 9)/(x - 3) = lim[x→3] (x + 3)(x - 3)/(x - 3) = lim[x→3] (x + 3) = 6

Method 3: Rationalize (for expressions with radicals)
lim[x→0] (√(x + 1) - 1)/x 
Multiply by (√(x + 1) + 1)/(√(x + 1) + 1):
= lim[x→0] ((x + 1) - 1)/(x(√(x + 1) + 1)) = lim[x→0] x/(x(√(x + 1) + 1)) = 1/2

Special Limits

lim[x→0] (sin x)/x = 1
lim[x→0] (1 - cos x)/x = 0
lim[x→∞] (1 + 1/x)ˣ = e
lim[x→0] (1 + x)^(1/x) = e

These are fundamental limits used in calculus

Continuity

f(x) is continuous at x = c if:
1. f(c) exists
2. lim[x→c] f(x) exists  
3. lim[x→c] f(x) = f(c)

Types of discontinuities:
• Removable: hole in graph, limit exists
• Jump: left and right limits differ
• Infinite: vertical asymptote

Example: f(x) = (x² - 4)/(x - 2)
At x = 2: function undefined (hole)
lim[x→2] f(x) = 4, but f(2) doesn't exist
Removable discontinuity

Mathematical Modeling

Exponential Models

Growth: P(t) = P₀e^(rt) where r > 0
Decay: P(t) = P₀e^(-rt) where r > 0

Population Growth:
If population doubles every 10 years:
P(t) = P₀ · 2^(t/10)

Radioactive Decay:
Half-life model: N(t) = N₀ · (1/2)^(t/h)
where h is half-life

Logarithmic Models

Logarithmic growth: y = a + b ln(x)
Common in:
• Learning curves
• Diminishing returns
• Psychological responses (Weber-Fechner law)

Example: Decibel scale
dB = 10 log₁₀(I/I₀)
where I is intensity, I₀ is reference intensity

Trigonometric Models

Periodic phenomena: y = A sin(Bt + C) + D

Applications:
• Sound waves: amplitude, frequency, phase
• Seasonal temperature variations
• Tidal patterns  
• Economic cycles
• Biorhythms

Example: Average monthly temperature
T(t) = 15 sin(π(t - 4)/6) + 20
where t is month (January = 1)

Logistic Models

Limited growth: P(t) = L/(1 + ae^(-bt))
where L is carrying capacity

Combines exponential growth (early stages) with 
leveling off (approaching limit)

Applications:
• Population growth with limited resources
• Technology adoption
• Spread of diseases
• Market penetration

Practice Problems

Functions

1. Given f(x) = x² - 3x + 2 and g(x) = 2x + 1, find:
   a) (f ∘ g)(x)
   b) (g ∘ f)(x)
   c) f⁻¹(x)

2. Find the domain of h(x) = √(x - 2)/(x² - 9)

3. Analyze the rational function R(x) = (x² - 1)/(x² - 4x + 3):
   Find vertical asymptotes, horizontal asymptotes, and holes.

Exponential and Logarithmic

1. Solve: 3^(2x-1) = 27^(x+2)

2. If ln(a) = 2 and ln(b) = 3, find ln(a²√b)

3. A radioactive substance decays according to A(t) = A₀e^(-0.05t).
   How long until half the substance remains?

Trigonometry

1. Find exact values:
   a) sin(7π/6)
   b) cos(5π/4)  
   c) tan(-π/3)

2. Verify the identity: (1 + cos θ)/(1 - cos θ) = (1 + sin θ)²/sin² θ

3. Solve: 2sin²x - 3sin x + 1 = 0 for 0 ≤ x < 2π

Sequences and Series

1. Find the 15th term of the arithmetic sequence: 4, 7, 10, 13, ...

2. Find the sum of the geometric series: 3 + 1 + 1/3 + 1/9 + ...

3. A ball is dropped from 20 feet. Each bounce reaches 3/4 the previous height.
   What is the total distance traveled?

Limits

1. Evaluate: lim[x→3] (x² - 9)/(x² - 5x + 6)

2. Evaluate: lim[x→0] (sin 3x)/x

3. Find: lim[x→∞] (2x³ - x + 1)/(x³ + 2x² - 5)

Answer Key

Functions

1. a) (f ∘ g)(x) = (2x + 1)² - 3(2x + 1) + 2 = 4x² - 4x
   b) (g ∘ f)(x) = 2(x² - 3x + 2) + 1 = 2x² - 6x + 5
   c) f⁻¹(x) = (3 ± √(4x - 7))/2

2. Domain: x ≥ 2 and x ≠ ±3, so [2, ∞) ∩ (-∞,-3) ∪ (-3,3) ∪ (3,∞) = [2,3) ∪ (3,∞)

3. R(x) = (x-1)(x+1)/((x-1)(x-3)) = (x+1)/(x-3) for x ≠ 1
   Vertical asymptote: x = 3
   Horizontal asymptote: y = 1  
   Hole: (1, -1/2)

Exponential and Logarithmic

1. 3^(2x-1) = (3³)^(x+2) = 3^(3x+6)
   2x - 1 = 3x + 6
   x = -7

2. ln(a²√b) = ln(a²) + ln(b^(1/2)) = 2ln(a) + (1/2)ln(b) = 2(2) + (1/2)(3) = 5.5

3. 0.5A₀ = A₀e^(-0.05t)
   0.5 = e^(-0.05t)
   ln(0.5) = -0.05t
   t = ln(0.5)/(-0.05) ≈ 13.86 time units

Trigonometry

1. a) sin(7π/6) = -1/2
   b) cos(5π/4) = -√2/2
   c) tan(-π/3) = -√3

2. [Identity verification by algebraic manipulation]

3. Let u = sin x: 2u² - 3u + 1 = 0
   (2u - 1)(u - 1) = 0
   u = 1/2 or u = 1
   sin x = 1/2: x = π/6, 5π/6
   sin x = 1: x = π/2

Sequences and Series

1. aₙ = 4 + (n-1)3 = 3n + 1
   a₁₅ = 3(15) + 1 = 46

2. S∞ = 3/(1-1/3) = 3/(2/3) = 9/2 = 4.5

3. Down: 20 + 20(3/4) + 20(3/4)² + ... = 20/(1-3/4) = 80
   Up: 20(3/4) + 20(3/4)² + ... = 60
   Total: 80 + 60 = 140 feet

Limits

1. lim[x→3] (x-3)(x+3)/((x-2)(x-3)) = lim[x→3] (x+3)/(x-2) = 6/1 = 6

2. lim[x→0] (sin 3x)/x = 3 · lim[x→0] sin(3x)/(3x) = 3 · 1 = 3

3. lim[x→∞] (2x³ - x + 1)/(x³ + 2x² - 5) = 2/1 = 2

Quick Reference

Function Operations

(f ∘ g)(x) = f(g(x))
f⁻¹ exists if f is one-to-one
Horizontal line test for one-to-one

Exponential/Logarithmic

logₐ(xy) = logₐ(x) + logₐ(y)
logₐ(xⁿ) = n logₐ(x)
Change of base: logₐ(x) = ln(x)/ln(a)

Trigonometry

sin²θ + cos²θ = 1
sin(A ± B) = sin A cos B ± cos A sin B
Period of sin, cos: 2π; Period of tan: π

Sequences/Series

Arithmetic: aₙ = a₁ + (n-1)d; Sₙ = n(a₁ + aₙ)/2
Geometric: aₙ = a₁rⁿ⁻¹; Sₙ = a₁(1-rⁿ)/(1-r)
Infinite geometric: S∞ = a₁/(1-r) if |r| < 1

Limits

lim[x→c] f(x) = L if f(x) → L as x → c
lim[x→0] (sin x)/x = 1
Continuity: f(c) exists, lim exists, lim = f(c)