Pre-Calculus Master Quiz

Unit 1 · Functions

Domain, Range, Composition & Inverse

Domain of f/g : exclude values where g(x) = 0 Domain of sqrt(f) : set f(x) >= 0 (f°g)(x) = f(g(x)) [apply g first, then f] Inverse : swap x and y, solve for y One-to-one ↔ passes Horizontal Line Test

★ For f(x) = sqrt(3x-6)/(x-5): need 3x-6 ≥ 0 AND x ≠ 5.

Example

f(x) = 2x+1, g(x) = x²−3. Find (f°g)(−2).

g(−2) = 4−3 = 1 → f(1) = 3 ✓

Unit 2 · Polynomials

Remainder Theorem · Factor Theorem · Rational Roots

Remainder Theorem : f(a) = remainder when f(x) ÷ (x−a) Factor Theorem : (x−a) is a factor ↔ f(a) = 0 Rational Root : p/q, where p | a&sub0;, q | a⊂n End Behavior : determined by leading term a⊂n·x⊃n

★ Even degree + negative leading coeff → both ends → −∞.

Example

f(x) = 2x³−x²−7x+6. Is x = −2 a zero?

f(−2) = −16−4+14+6 = 0 ✓ Yes!

Unit 3 · Exponentials & Logarithms

Log Laws & Change of Base

log_b(xy) = log_b(x) + log_b(y) log_b(x/y) = log_b(x) − log_b(y) log_b(x^n) = n · log_b(x) Change of base : log_b(x) = ln(x)/ln(b) b^x = y ↔ log_b(y) = x ln(e^x) = x, e^(ln x) = x

★ log is undefined for non-positive arguments. Always check domain!

Example

Solve: log⊂2;(x−3) + log⊂2;(x+1) = 5

(x−3)(x+1) = 32 → x²−2x−35=0 → x=7 ✓

Unit 4 · Trigonometry

Unit Circle, Identities, Graphs & Laws

sin²θ + cos²θ = 1 tanθ = sinθ/cosθ sin(2θ) = 2 sinθ cosθ cos(2θ) = 2cos²θ−1 = 1−2sin²θ Period of A·sin(Bx+C) : T = 2π/|B| QIII: sin(−), cos(−); Reference angle for 225° = 45° Law of Cosines : c² = a²+b²−2ab·cosC

★ cos(2θ) = 2cos²θ−1 is the most useful double-angle form when cosθ is given.

Example

cosθ = −3/5, find cos(2θ).

2(9/25)−1 = 18/25−25/25 = −7/25 ✓

Unit 5 · Conic Sections

Circle, Ellipse, Hyperbola, Parabola

Circle : (x−h)²+(y−k)² = r² Ellipse : x²/a²+y²/b² = 1, a>b, c²=a²−b², foci (±c,0) Hyperbola : x²/a²−y²/b² = 1, c²=a²+b², asymptotes y=±(b/a)x Parabola : (x−h)² = 4p(y−k) [vertical opening]

★ For ellipse x²/25+y²/9=1: a²=25, b²=9, c²=25−9=16, c=4, foci at (±4,0).

Example

x²+y²−6x+8y=0 → complete the square.

(x−3)²+(y+4)²=25, r=5 ✓

Unit 6 · Sequences & Series

Arithmetic & Geometric Sequences

Arithmetic : a_n = a_1+(n−1)d, S_n = n/2·(a_1+a_n) Geometric : a_n = a_1·r^(n−1), S_n = a_1(1−r^n)/(1−r) Infinite Geo: S_∞ = a_1/(1−r), |r| < 1 required

★ Always check |r| < 1 before using the infinite sum formula.

Example

Arithmetic: a⊂1;=3, d=4, n=20. Find S⊂20;.

a⊂20;=79, S⊂20;=10×82=820 ✓

Unit 7 · Matrices & Vectors

Determinant, Dot Product & Angles

det[[a,b],[c,d]] = ad − bc Dot product : u·v = u⊂1;v⊂1;+u⊂2;v⊂2; cosθ = (u·v)/(|u||v|) Perpendicular ↔ u·v = 0

★ If dot product = 0, vectors are perpendicular (90°) regardless of their lengths.

Example

u=⟨3,4⟩, v=⟨4,−3⟩. u·v = 12−12 = 0 → 90°

Vectors are perpendicular ✓

Unit 8 · Complex Numbers

Operations, Modulus & Powers of i

i=√(−1), i²=−1, i³=−i, i&sup4;=1 [cycle of 4] (a+bi)(c+di) = (ac−bd)+(ad+bc)i (a+bi)(a−bi) = a²+b² [no imaginary part!] i^n : find n mod 4, then i¹=i, i²=−1, i³=−i, i&sup4;=1

★ (a+bi)(a−bi) = a²+b² always gives a real number.

Example

(3+2i)(3−2i) = 3²+2² = 9+4 = 13 ✓

i³&sup7;: 37 mod 4 = 1 → i¹ = i ✓

Pre-CalculusMaster Quiz

Key Concepts & Formulas to Memorize

Pre-Calculus
Master Quiz