Advanced Algebra

Functions

A function maps each input to exactly one output. Written f(x).

• f(x) = 3x − 2: f(5) = 15 − 2 = 13
• Domain: set of valid inputs   Range: set of resulting outputs
• Composition: f(g(x)) — apply g first, then f to the result.
• Inverse: swap x and y, solve for y to find f⁻¹(x).

Quadratics

Standard form: ax² + bx + c = 0

• Quadratic formula: x = [−b ± √(b²−4ac)] / 2a
• Discriminant (b²−4ac): positive = 2 real roots; zero = 1 real root; negative = no real roots.
• Vertex: x = −b/2a; substitute back to find y.
• Factoring: x²−5x+6 = (x−2)(x−3) → roots are x=2, x=3.

Sequences

• Arithmetic: constant difference (d). aₙ = a₁ + (n−1)d   Sum = n/2(a₁+aₙ)
• Geometric: constant ratio (r). aₙ = a₁·rⁿ⁻¹

Example: 3, 6, 12, 24 … (geometric, r=2). 10th term = 3×2⁹ = 3×512 = 1536.