Pre-Calculus Mastery Series

Pre-Calculus
Master Quiz

20 Essential Problems · All Major Topics · Exam Style

📐 Functions 📈 Polynomials 🔁 Trigonometry 📊 Logarithms 🎯 Conics ∑ Sequences
TIME REMAINING
40:00
ANSWERED
0 / 20
📚 Concept Review
Tap each topic to expand — memorize key formulas before starting
Unit 1

Functions & Their Properties

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Domain: all valid input values of x. Range: all output values of f(x).
Composition: (f∘g)(x) = f(g(x)) — apply g first, then f.
Inverse: f⁻¹ exists only when f is one-to-one (passes horizontal line test).
Vertical line test → determines if a relation is a function
Horizontal line test → determines if inverse is also a function
📌 Memorize
f(f⁻¹(x)) = x  |  f⁻¹(f(x)) = x
Even function: f(-x) = f(x)  |  Odd: f(-x) = -f(x)
Example: If f(x) = 2x+1, find f⁻¹(x).
Swap x and y: x = 2y+1 → y = (x−1)/2. So f⁻¹(x) = (x−1)/2
Unit 2

Polynomials & Rational Functions

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Remainder Theorem: dividing p(x) by (x−c) gives remainder p(c).
Factor Theorem: (x−c) is a factor of p(x) if and only if p(c) = 0.
Rational Root Theorem: possible rational roots = ±(factors of constant term)/(factors of leading coefficient).
Degree n polynomial has exactly n complex roots (counting multiplicity)
End behavior: determined by leading term aₙxⁿ
📌 Memorize
Vertical asymptote: set denominator = 0 (not cancelled)
Horizontal asymptote: compare degrees of num. vs denom.
Example: f(x) = (2x²+1)/(x²−4). HA: y = 2 (equal degrees → ratio of leading coeff.)
Unit 3

Exponential & Logarithmic Functions

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log_b(x) = y ↔ bʸ = x
Natural log: ln(x) = log_e(x).   Common log: log(x) = log₁₀(x).
Product: log(MN) = log M + log N
Quotient: log(M/N) = log M − log N
Power: log(Mⁿ) = n·log M
Change of base: log_b(x) = ln(x)/ln(b)
📌 Memorize
e ≈ 2.718  |  ln(e) = 1  |  ln(1) = 0
log_b(b) = 1  |  log_b(1) = 0  |  b^(log_b x) = x
Example: Solve 2^x = 32.   2^x = 2^5, so x = 5.
Unit 4

Trigonometry

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Unit circle: (cos θ, sin θ) for angle θ. Radius = 1.
Key angles (radians): 0, π/6, π/4, π/3, π/2, π, 3π/2, 2π.
sin²θ + cos²θ = 1  |  1+tan²θ = sec²θ  |  1+cot²θ = csc²θ
sin(A±B) = sinA·cosB ± cosA·sinB
cos(A±B) = cosA·cosB ∓ sinA·sinB
sin(2A) = 2sinA·cosA  |  cos(2A) = cos²A − sin²A
📌 Memorize — ASTC Rule (All Students Take Calculus)
Q1: All positive  |  Q2: Sin+  |  Q3: Tan+  |  Q4: Cos+
Period of sin/cos = 2π  |  Period of tan = π
Example: sin(π/6) = 1/2, cos(π/3) = 1/2, tan(π/4) = 1
Unit 5

Conic Sections

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Circle: (x−h)²+(y−k)² = r²
Parabola: y = a(x−h)² + k (vertex form)
Ellipse: (x−h)²/a² + (y−k)²/b² = 1, where a > b > 0
Hyperbola: (x−h)²/a² − (y−k)²/b² = 1
📌 Memorize
Ellipse: c² = a²−b²  |  Hyperbola: c² = a²+b²
Eccentricity e: circle=0, ellipse=0~1, parabola=1, hyperbola>1
Example: x²/25 + y²/9 = 1. Here a=5, b=3, c=√(25−9)=4. Foci at (±4, 0).
Unit 6

Sequences & Series

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Arithmetic: aₙ = a₁ + (n−1)d  |  S_n = n(a₁+aₙ)/2
Geometric: aₙ = a₁·rⁿ⁻¹  |  S_n = a₁(1−rⁿ)/(1−r), r≠1
Infinite geometric series (|r| < 1): S = a₁/(1−r)
📌 Memorize
Binomial Theorem: (a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ
C(n,k) = n! / (k!(n−k)!)
Example: Sum of infinite series: 1 + 1/2 + 1/4 + … = 1/(1−1/2) = 2
📝 Practice Problems
20 Multiple Choice · Select one answer per question
01 Functions ⭐⭐
Let f(x) = 3x − 2 and g(x) = x² + 1. What is (f∘g)(2)?
1Compute g(2) first: g(2) = 2² + 1 = 5
2Then apply f to that result: f(g(2)) = f(5) = 3(5) − 2 = 13
✓Answer: A — 13
02 Functions ⭐⭐
Which of the following functions is an even function?
1Even function condition: f(−x) = f(x) for all x.
2Test B: f(−x) = (−x)² + 3 = x² + 3 = f(x). ✓ Even!
3A is odd (x³+x), C is neither, D is neither.
✓Answer: B — f(x) = x² + 3
03 Polynomials ⭐⭐⭐
When p(x) = x³ − 4x² + 5x − 2 is divided by (x − 2), what is the remainder?
1Remainder Theorem: remainder = p(2)
2p(2) = 8 − 16 + 10 − 2 = 0
✓Answer: A — 0 (so (x−2) is a factor!)
04 Rational Functions ⭐⭐⭐
What is the horizontal asymptote of f(x) = (3x² − 1) / (x² + 5)?
1Numerator degree = 2, Denominator degree = 2. Equal degrees.
2HA = ratio of leading coefficients: 3/1 = 3
✓Answer: B — y = 3
05 Exponential Functions ⭐⭐
Solve for x: 4^(x+1) = 8^x
1Write both sides as powers of 2: 2^(2(x+1)) = 2^(3x)
2Set exponents equal: 2x + 2 = 3x → x = 2
✓Answer: B — x = 2
06 Logarithms ⭐⭐⭐
Simplify: log₂(32) − log₂(4)
1Quotient rule: log₂(32/4) = log₂(8)
2log₂(8) = log₂(2³) = 3
✓Answer: A — 3
07 Trigonometry ⭐⭐
If sin θ = 3/5 and θ is in the first quadrant, what is cos θ?
1Use Pythagorean identity: cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25
2Q1 → cosθ > 0: cos θ = 4/5
✓Answer: A — 4/5
08 Trig Identities ⭐⭐⭐
Which expression is equal to sin(2θ)?
1Double angle formula: sin(2θ) = 2sinθ·cosθ
2Note: C is cos(2θ), and D = 1 (Pythagorean identity).
✓Answer: B — 2sinθ · cosθ
09 Inverse Trig ⭐⭐⭐
What is the value of arcsin(−1/2) in radians (principal value)?
1arcsin has range [−π/2, π/2].
2sin(π/6) = 1/2, so sin(−π/6) = −1/2.
✓Answer: A — −π/6
10 Conic Sections ⭐⭐⭐
The equation x²/16 + y²/9 = 1 represents an ellipse. What is the distance between the two foci?
1a² = 16, b² = 9. For ellipse: c² = a² − b² = 16 − 9 = 7
2c = √7. Distance between foci = 2c = 2√7
✓Answer: A — 2√7
11 Conic Sections ⭐⭐
What are the coordinates of the vertex of the parabola y = 2(x − 3)² + 5?
1Vertex form: y = a(x−h)² + k → vertex = (h, k)
2Here h = 3, k = 5 → vertex = (3, 5)
✓Answer: A — (3, 5)
12 Sequences ⭐⭐
Find the 10th term of the arithmetic sequence: 3, 7, 11, 15, …
1a₁ = 3, common difference d = 4
2a₁₀ = 3 + (10−1)×4 = 3 + 36 = 39
✓Answer: A — 39
13 Series ⭐⭐⭐
What is the sum of the infinite geometric series 6 + 2 + 2/3 + …?
1a₁ = 6, r = 2/6 = 1/3. Since |1/3| < 1, series converges.
2S = 6 / (1 − 1/3) = 6 / (2/3) = 9
✓Answer: A — 9
14 Binomial Theorem ⭐⭐⭐⭐
What is the coefficient of x² in the expansion of (x + 2)⁴?
1x² term: k=2 in C(4,2)·x²·2²
2C(4,2) = 6, 2² = 4 → coefficient = 6 × 4 = 24
✓Answer: A — 24
15 Logarithms ⭐⭐⭐
Solve for x: log₃(x − 2) + log₃(x + 2) = 2
1Product rule: log₃[(x−2)(x+2)] = 2
2Convert: (x−2)(x+2) = 3² = 9
3x² − 4 = 9 → x² = 13 → x = √13 (x > 2 required)
✓Answer: A — x = √13
16 Trig Equations ⭐⭐⭐
How many solutions does 2sin(x) − 1 = 0 have on [0, 2π)?
1sin(x) = 1/2 → x = π/6 and x = 5π/6
2Both values lie in [0, 2π). So there are 2 solutions.
✓Answer: B — 2
17 Functions — Inverse ⭐⭐⭐
If f(x) = (x + 4) / (x − 1), what is f⁻¹(x)?
1Set y = (x+4)/(x−1), swap x and y: x = (y+4)/(y−1)
2Solve for y: x(y−1) = y+4 → xy − x = y + 4 → y(x−1) = x+4 → y = (x+4)/(x−1)
✓Answer: A — this function is self-inverse (its own inverse)!
18 Polynomials — Roots ⭐⭐⭐⭐
A polynomial with real coefficients has roots 3 and 2 + i. What must also be a root?
1Complex Conjugate Root Theorem: if a+bi is a root and coefficients are real, then a−bi is also a root.
2Root is 2+i → conjugate must be 2−i
✓Answer: A — 2 − i
19 Exponential Growth ⭐⭐⭐⭐
A population doubles every 5 years. If the population is 1,000 now, which expression gives the population after t years?
1Doubling time model: P(t) = P₀ · 2^(t/T) where T = doubling time = 5
2Check: at t=5, P = 1000·2^(5/5) = 1000·2 = 2000 ✓
✓Answer: A — 1000 · 2^(t/5)
20 Trig — Law of Cosines ⭐⭐⭐⭐
In triangle ABC, a = 5, b = 7, C = 60°. What is the length of side c?
1Law of Cosines: c² = a² + b² − 2ab·cos C
2c² = 25 + 49 − 2(5)(7)cos(60°) = 74 − 70·(1/2) = 74 − 35 = 39
3c = √39
✓Answer: A — √39
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