📚 Core Concepts Review
Study each topic before starting — tap a tab to switch
f Functions & Composition
A function maps each input to exactly one output. Domain = valid inputs; Range = all outputs.
📌 Memorize
- Domain: denominator ≠ 0 and radicand ≥ 0
- Composition: (f ∘ g)(x) = f(g(x)) — apply inner first
- Vertical Line Test: function if no vertical line crosses twice
- Inverse f⁻¹: swap x and y; domain of f = range of f⁻¹
✦ Quick Example
f(x) = 2x + 1, g(x) = x². Find (f ∘ g)(3).
→ g(3) = 9; f(9) = 2(9) + 1 = 19
Pn Polynomial Functions
Form: anxn + … + a₁x + a₀. End behavior is controlled by the leading term.
📌 Memorize
- Factor Theorem: f(c) = 0 ⇔ (x − c) is a factor
- Degree-n polynomial has at most n real zeros
✦ Quick Example
End behavior of f(x) = −3x⁴ + 2x² − 1?
→ Even (4), negative leading coeff: f(x) → −∞ as x → ±∞
p/q Rational Functions
Ratio of two polynomials. Asymptotes control the graph’s long-run behavior.
📌 Asymptote Rules
- Vertical Asymptote: set denominator = 0 (after canceling)
- deg(num) < deg(den) → HA: y = 0
- deg(num) = deg(den) → HA: y = ratio of leading coefficients
- deg(num) > deg(den) → No HA (oblique asymptote)
✦ Quick Example
HA of f(x) = (3x² + 2) / (x² − 5)?
→ Equal degrees; HA: y = 3/1 = 3
ex Exponential & Logarithmic
These are inverse operations: ax = y ⇔ loga(y) = x.
📌 Log Laws
- Product: loga(MN) = logaM + logaN
- Quotient: loga(M/N) = logaM − logaN
- Power: loga(Mn) = n·logaM
- Change of base: loga(x) = ln(x)/ln(a)
- Compound interest: A = P(1 + r)t (annual)
✦ Quick Example
Solve: 4(x−1) = 64
→ 64 = 4³, so x−1=3, thus x = 4
θ Trigonometry
📌 Key Unit Circle Values
- sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3
- sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
- sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
- arctan(1) = π/4; arcsin(1/2) = π/6
✦ Quick Example
Find cos(150°)
→ Q2, ref = 30°, cos negative → −√3/2
Σ Sequences & Series
✦ Quick Example
20th term of 5, 9, 13, 17, …
→ a₁=5, d=4; aₙ=5+(19)(4)=5+76= 81
◯ Conic Sections
📌 Key Facts
- Ellipse: if a > b, major axis horizontal; vertices at (±a, 0)
- Parabola y = a(x−h)²+k: vertex (h, k); opens up if a>0
- Circle: center (h, k), radius r
✦ Quick Example
Vertex of y = −2(x−3)² + 4?
→ Vertex form (h,k): vertex = (3, 4)
i Complex Numbers
📌 Memorize
- i = √(−1), i² = −1, i³ = −i, i⁴ = 1
- Conjugate of (a + bi) is (a − bi)
- (a + bi)(a − bi) = a² + b² (always real!)
- Modulus: |a + bi| = √(a² + b²)
✦ Quick Example
Compute (3 + 2i)(3 − 2i)
→ a² + b² = 3² + 2² = 9 + 4 = 13