Pre-Calculus Master Series

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Core Concepts & Key Formulas

UNIT 1 Functions & Their Properties

A function maps each input x to exactly one output f(x).
  • Domain: all valid x-values  |  Range: all resulting y-values
  • Even: f(−x) = f(x) — symmetric about y-axis
  • Odd: f(−x) = −f(x) — symmetric about origin
  • Composition: (f ∘ g)(x) = f(g(x)) — apply g first, then f
  • Inverse exists iff f is one-to-one; f(f¹(x)) = x
Example

If f(x) = 2x + 3 and g(x) = x², find (f ∘ g)(2).

g(2) = 4, then f(4) = 2(4) + 3 = 11

UNIT 2 Polynomial Functions

  • Degree-n polynomial: at most n real zeros, at most n−1 turning points
  • Remainder Theorem: dividing f(x) by (x−c) gives remainder f(c)
  • Factor Theorem: (x−c) is a factor if and only if f(c) = 0
  • Complex roots of real polynomials come in conjugate pairs
End behavior (leading term axⁿ): a > 0, n even → ↑ both ends a > 0, n odd → ↓ left, ↑ right a < 0 reverses direction
Example

Remainder when P(x) = x³ − 4x + 1 is divided by (x − 2).

P(2) = 8 − 8 + 1 = 1

UNIT 3 Rational Functions & Asymptotes

  • Vertical asymptote x = a: denominator = 0, numerator ≠ 0 at x = a
  • Horizontal asymptote: deg(num) < deg(den) → y = 0; equal degrees → y = ratio of leading coefficients; deg(num) > deg(den) → none
  • Hole: factor cancels from both numerator and denominator
  • Oblique asymptote: when deg(num) = deg(den) + 1, use long division
Example

H.A. of f(x) = (3x² − 1) / (2x² + 5).

Same degree → H.A. is y = 3/2

UNIT 4 Exponential & Logarithmic Functions

  • logₛ(x) = y ↔ bʏ = x  (b > 0, b ≠ 1, x > 0)
  • Product: log(MN) = log(M) + log(N)
  • Quotient: log(M/N) = log(M) − log(N)
  • Power: log(Mⁿ) = n · log(M)
  • Change of base: logₛ(x) = ln(x)/ln(b)
Compound Interest: A = P(1 + r/n)^(nt) Continuous: A = Pe^(rt)
Example

Solve: log₂(x) + log₂(x − 2) = 3

log₂[x(x−2)] = 3 → x(x−2) = 8 → x²−2x−8 = 0 → (x−4)(x+2) = 0 → x = 4

UNIT 5 Trigonometric Functions

  • sin²θ + cos²θ = 1  |  1 + tan²θ = sec²θ
  • sin(A ± B) = sinA cosB ± cosA sinB
  • cos(A ± B) = cosA cosB &mnsp; sinA sinB
  • Double angle: sin2θ = 2sinθcosθ  |  cos2θ = cos²θ − sin²θ
  • Law of Sines: a/sinA = b/sinB = c/sinC
  • Law of Cosines: c² = a² + b² − 2ab cosC
y = A sin(Bx + C) + D: Amplitude = |A| Period = 2π/|B| Phase shift = −C/B
Example

Period of y = 3 sin(2x − π/4) + 1.

Period = 2π/|B| = 2π/2 = π

UNIT 6 Inverse Trig & Polar Coordinates

  • arcsin: domain [−1, 1], range [−π/2, π/2]
  • arccos: domain [−1, 1], range [0, π]
  • arctan: domain ℝ, range (−π/2, π/2)
  • Polar: x = r cosθ, y = r sinθ, r = √(x²+y²)
Example

Evaluate arctan(√3).

tan(π/3) = √3 and π/3 ∈ (−π/2, π/2), so arctan(√3) = π/3

UNIT 7 Conic Sections

Circle: (x−h)² + (y−k)² = r² Ellipse: (x−h)²/a² + (y−k)²/b² = 1 (a > b, horiz. major) Hyperbola: (x−h)²/a² − (y−k)²/b² = 1 Parabola: y−k = (1/4p)(x−h)²
  • Ellipse foci: c² = a² − b², foci at (h ± c, k)
  • Hyperbola foci: c² = a² + b²
  • Asymptotes of hyperbola: y − k = ±(b/a)(x − h)
Example

Identify: 4x² + 9y² = 36

Divide by 36: x²/9 + y²/4 = 1 → Ellipse, a = 3, b = 2

UNIT 8 Vectors

  • Magnitude: |v| = √(a² + b²)
  • Unit vector: v̂ = v / |v|
  • Dot product: u · v = u₁v₁ + u₂v₂ = |u||v| cosθ
  • Perpendicular: u · v = 0
  • Angle between vectors: cosθ = (u · v) / (|u||v|)
Example

Angle between u = ⟨1, 0⟩ and v = ⟨1, 1⟩.

cosθ = 1/(1·√2) = 1/√2 → θ = 45° = π/4

UNIT 9 Sequences, Series & Binomial Theorem

Arithmetic: aₙ = a₁ + (n−1)d Sₙ = n(a₁+aₙ)/2 Geometric: aₙ = a₁·r^(n−1) Sₙ = a₁(1−rⁿ)/(1−r) Inf. Geom.: S = a₁/(1−r) if |r| < 1 Binomial: (a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ
Example

Sum of 8 + 4 + 2 + 1 + …

a₁ = 8, r = 1/2. S = 8/(1−1/2) = 16

UNIT 10 Introduction to Limits

  • lim[x→a] f(x) = L: f(x) approaches L as x approaches a
  • Two-sided limit exists only when left and right limits are equal
  • Direct substitution works when f is continuous at a
  • 0/0 form: factor, rationalize, or simplify before substituting
Fundamental: lim[x→0] (sin x)/x = 1
Example

Find: lim[x→2] (x² − 4) / (x − 2)

Factor: (x+2)(x−2)/(x−2) = x+2 → lim = 2+2 = 4
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Pre-Calculus Core Concepts — 20 Questions

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