📚 Key Concepts & Formulas to Memorize

Unit 1

Functions & Domain

Dom(√f) : f(x) ≥ 0
Dom(1/f) : f(x) ≠ 0

Find where the expression is defined. Combine both restrictions when both apply.

Unit 2

Composition & Inverse

(f∘g)(x) = f(g(x))
f⁻¹: swap x↔y, solve y

Always apply inner function first. Verify: f(f⁻¹(x)) = x.

Unit 3

Polynomial Zeros

Vieta's: sum = −b/a
product = c/a (quadratic)

For xⁿ + bxⁿ⁻¹ + …, sum of roots = −b.

Unit 4

Rational Functions

VA: denominator = 0
Hole: common factor cancels

Always factor and cancel common factors first before finding asymptotes.

Unit 5

Exponential Laws

aˣ = aʸ ⟹ x = y
8 = 2³, 9 = 3², 27 = 3³

Convert all bases to the same prime base to equate exponents.

Unit 6

Logarithm Rules

log(ab) = log a + log b
log(aⁿ) = n·log a
logₐa = 1

Combine logs before solving. Always check domain (argument > 0).

Unit 7

Trig Exact Values

sin30°=½, cos60°=½
sin45°=cos45°=√2/2
sin60°=√3/2

Use sum/difference formulas for non-standard angles like 75°, 15°.

Unit 8

Trig Equations

sin θ = k ⟹
θ = arcsin(k) + 2πn
or π − arcsin(k) + 2πn

Factor trig equations like quadratics. Find all solutions in [0, 2π).

Unit 9

Law of Cosines

c² = a² + b² − 2ab·cos C

Use when given SAS or SSS. Law of Sines for AAS or ASA.

Unit 10

Sequences & Series

Arithmetic: aₙ = a₁+(n−1)d
Sₙ = n(a₁+aₙ)/2
Geometric: Sₙ = a₁(rⁿ−1)/(r−1)

Identify type first: constant difference → arithmetic; constant ratio → geometric.

Unit 11

Binomial Theorem

(a+b)ⁿ = Σ C(n,k)aⁿ⁻ᵏbᵏ
C(n,k) = n!/(k!(n−k)!)

The term with bᵏ is the (k+1)th term. Match exponent of desired variable.

Unit 12

Conics — Parabola

y² = 4px → focus (p,0)
x² = 4py → focus (0,p)

p > 0 opens right/up; p < 0 opens left/down. Directrix is −p from vertex.

Unit 13

Conics — Ellipse

x²/a²+y²/b²=1, a>b
c² = a²−b²
foci: (±c, 0)

The larger denominator determines the major axis orientation.

Unit 14

Polar ↔ Rectangular

x = r·cos θ, y = r·sin θ
r² = x²+y², tan θ = y/x

Memorize: (r, θ) ↔ (r cos θ, r sin θ). Watch quadrant for θ.

Unit 15

Limits (Intro)

lim(x→a) [0/0]: factor & cancel
lim(x→∞) rational: leading terms

For 0/0 indeterminate forms, factor numerator/denominator and simplify.

Unit 16

Complex Numbers

(a+bi)(c+di) = (ac−bd)+(ad+bc)i
i² = −1

Use FOIL and replace i² with −1. Conjugate for division.

Unit 17

Matrices

det|a b; c d| = ad − bc
A⁻¹ = (1/det)·adj(A)

Matrix multiplication is not commutative. AB ≠ BA in general.

Unit 18

Vectors & Dot Product

u·v = u₁v₁+u₂v₂+u₃v₃
cos θ = (u·v)/(|u||v|)

Perpendicular ⟺ dot product = 0. Parallel ⟺ one is scalar multiple.

💡 Worked Examples

📘 Example A — Domain of a Function

Find the domain of \( f(x) = \dfrac{\sqrt{x-3}}{x-5} \)

Radicand must be non-negative: \(x - 3 \geq 0 \Rightarrow x \geq 3\)
Denominator cannot be zero: \(x - 5 \neq 0 \Rightarrow x \neq 5\)
Combine: \(x \geq 3\) AND \(x \neq 5\)

✅ Domain: \([3, 5) \cup (5, \infty)\)

📘 Example B — Exponential Equation

Solve: \(2^{2x-1} = 8^{x-2}\)

Rewrite with same base: \(8 = 2^3\), so \(8^{x-2} = 2^{3(x-2)} = 2^{3x-6}\)
Equate exponents: \(2x - 1 = 3x - 6\)
Solve: \(-1 + 6 = 3x - 2x \Rightarrow x = 5\)

✅ Answer: \(x = 5\)

📘 Example C — Trig Addition Formula

Find the exact value of \(\sin 75°\)

Write \(75° = 45° + 30°\)
Apply: \(\sin(A+B) = \sin A \cos B + \cos A \sin B\)
\(= \tfrac{\sqrt{2}}{2}\cdot\tfrac{\sqrt{3}}{2} + \tfrac{\sqrt{2}}{2}\cdot\tfrac{1}{2} = \tfrac{\sqrt{6}}{4} + \tfrac{\sqrt{2}}{4}\)

✅ Answer: \(\dfrac{\sqrt{6}+\sqrt{2}}{4}\)

Functions & Domain

What is the domain of \( f(x) = \dfrac{\sqrt{x-3}}{x-5} \)?

We need \(x - 3 \geq 0\) (radicand) giving \(x \geq 3\), and \(x - 5 \neq 0\) giving \(x \neq 5\). Combining: \([3, 5) \cup (5, \infty)\).

Composition of Functions

Given \(f(x) = 2x + 1\) and \(g(x) = x^2 - 3\), find \((f \circ g)(2)\).

\((f \circ g)(2) = f(g(2))\). First, \(g(2) = 4 - 3 = 1\). Then \(f(1) = 2(1) + 1 = 3\).

Inverse Functions

If \(f(x) = \dfrac{3x - 1}{x + 2}\), which of the following is \(f^{-1}(x)\)?

Set \(y = \frac{3x-1}{x+2}\). Then \(y(x+2) = 3x-1\), so \(xy + 2y = 3x - 1\), giving \(x(y-3) = -1 - 2y\), thus \(x = \frac{-(1+2y)}{y-3} = \frac{2y+1}{3-y}\). Replace \(y\) with \(x\): \(f^{-1}(x) = \dfrac{2x+1}{3-x}\).

Polynomial Functions

The polynomial \(p(x) = x^3 - 6x^2 + 11x - 6\) has three real roots. What is their sum?

By Vieta's formulas, the sum of roots of \(x^3 + bx^2 + \cdots\) is \(-b\). Here the coefficient of \(x^2\) is \(-6\), so sum \(= -(-6) = 6\). (The roots are \(1, 2, 3\) and \(1+2+3=6\).)

Rational Functions

Which best describes \(f(x) = \dfrac{x^2 - 4}{x^2 - x - 6}\)?

Factor: \(\dfrac{(x-2)(x+2)}{(x-3)(x+2)}\). The factor \((x+2)\) cancels, creating a hole at \(x = -2\). The remaining denominator \((x-3) = 0\) gives a vertical asymptote at \(x = 3\).

Exponential Equations

Solve for \(x\): \(2^{2x-1} = 8^{x-2}\)

Write \(8 = 2^3\), so \(8^{x-2} = 2^{3(x-2)} = 2^{3x-6}\). Setting exponents equal: \(2x - 1 = 3x - 6\), so \(x = 5\). Check: \(2^9 = 512 = 8^3\). ✓

Logarithmic Equations

Solve: \(\log_2 x + \log_2(x - 6) = 4\)

Combine: \(\log_2[x(x-6)] = 4\), so \(x(x-6) = 2^4 = 16\). This gives \(x^2 - 6x - 16 = 0\), factoring as \((x-8)(x+2) = 0\). So \(x = 8\) or \(x = -2\). Since \(\log_2(-2)\) is undefined, we reject \(x = -2\). Answer: \(x = 8\).

Trigonometry — Exact Values

What is the exact value of \(\sin 75°\)?

\(\sin 75° = \sin(45°+30°) = \sin 45°\cos 30° + \cos 45°\sin 30°\) \(= \tfrac{\sqrt{2}}{2}\cdot\tfrac{\sqrt{3}}{2} + \tfrac{\sqrt{2}}{2}\cdot\tfrac{1}{2} = \tfrac{\sqrt{6}}{4} + \tfrac{\sqrt{2}}{4} = \dfrac{\sqrt{6}+\sqrt{2}}{4}\).

Trigonometric Equations

Find all solutions of \(2\sin^2 x - \sin x - 1 = 0\) in \([0, 2\pi)\). How many solutions are there?

Factor: \((2\sin x + 1)(\sin x - 1) = 0\). So \(\sin x = -\tfrac{1}{2}\) giving \(x = \tfrac{7\pi}{6}, \tfrac{11\pi}{6}\); or \(\sin x = 1\) giving \(x = \tfrac{\pi}{2}\). Total: 3 solutions.

Law of Cosines

In triangle \(ABC\), \(a = 7\), \(b = 5\), and \(C = 60°\). Find \(c^2\).

Law of Cosines: \(c^2 = a^2 + b^2 - 2ab\cos C\). \(= 49 + 25 - 2(7)(5)\cos 60°\) \(= 74 - 70 \cdot \tfrac{1}{2} = 74 - 35 = 39\).

Arithmetic Sequences

An arithmetic sequence has first term \(a_1 = 3\) and common difference \(d = 4\). What is \(S_{10}\)?

\(a_{10} = 3 + (10-1)(4) = 3 + 36 = 39\). Then \(S_{10} = \frac{10}{2}(a_1 + a_{10}) = 5(3 + 39) = 5 \times 42 = 210\).

Geometric Series

A geometric sequence has \(a_1 = 2\) and common ratio \(r = 3\). Find \(S_5\).

\(S_5 = \dfrac{a_1(r^5 - 1)}{r - 1} = \dfrac{2(243 - 1)}{2} = \dfrac{2 \times 242}{2} = 242\).

Binomial Theorem

What is the coefficient of \(x^3\) in the expansion of \((x + 2)^5\)?

The term with \(x^3\) is \(\binom{5}{2} x^3 \cdot 2^2\). Here \(\binom{5}{2} = 10\) and \(2^2 = 4\). Coefficient \(= 10 \times 4 = 40\).

Polar Coordinates

Convert the polar point \(\left(3,\, \dfrac{\pi}{3}\right)\) to rectangular coordinates.

\(x = r\cos\theta = 3\cos\frac{\pi}{3} = 3 \cdot \frac{1}{2} = \frac{3}{2}\). \(y = r\sin\theta = 3\sin\frac{\pi}{3} = 3 \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}\). Answer: \(\left(\frac{3}{2}, \frac{3\sqrt{3}}{2}\right)\).

Conics — Parabola

For the parabola \(y^2 = 12x\), what is the focus?

Standard form \(y^2 = 4px\). Here \(4p = 12\), so \(p = 3\). For this orientation the focus is at \((p, 0) = (3, 0)\) and the directrix is \(x = -3\).

Conics — Ellipse

For the ellipse \(\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1\), where are the foci?

\(a^2 = 25, b^2 = 9\). So \(c^2 = a^2 - b^2 = 25 - 9 = 16\), giving \(c = 4\). Since \(a^2 > b^2\) along the \(x\)-axis, foci are at \((\pm 4, 0)\).

Limits

Evaluate: \(\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\)

The form \(\frac{0}{0}\) is indeterminate. Factor: \(\frac{(x-2)(x+2)}{x-2} = x + 2\) for \(x \neq 2\). Taking the limit: \(\lim_{x\to 2}(x+2) = 4\).

Complex Numbers

Simplify \((3 + 2i)(1 - 4i)\).

FOIL: \(3(1) + 3(-4i) + 2i(1) + 2i(-4i)\) \(= 3 - 12i + 2i - 8i^2\). Since \(i^2 = -1\): \(= 3 - 10i - 8(-1) = 3 - 10i + 8 = 11 - 10i\).

Matrices — Determinant

Find the determinant of \(A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\).

\(\det(A) = ad - bc = (2)(4) - (3)(1) = 8 - 3 = 5\).

Vectors — Dot Product

Given \(\mathbf{u} = \langle 2, -1, 3 \rangle\) and \(\mathbf{v} = \langle 4, 2, -1 \rangle\), what is \(\mathbf{u} \cdot \mathbf{v}\)?

\(\mathbf{u} \cdot \mathbf{v} = (2)(4) + (-1)(2) + (3)(-1) = 8 - 2 - 3 = 3\).

--%

SCORE

Correct ✓

Incorrect ✗

--:--

Time Used

📋 Complete Answer Key & Explanations

Precalculus Master Exam — 20 Questions

Name: _________________________ Date: _____________ Score: _____ / 20

1. [Functions] What is the domain of f(x) = √(x−3) / (x−5)?

(A) [3, ∞) (B) [3, 5) ∪ (5, ∞) (C) (5, ∞) (D) (−∞,5)∪(5,∞)

2. [Composition] Given f(x) = 2x+1 and g(x) = x²−3, find (f∘g)(2).

(A) 0 (B) 3 (C) 7 (D) 5

3. [Inverse] If f(x) = (3x−1)/(x+2), which is f⁻¹(x)?

(A) (x+2)/(3x−1) (B) (2x+1)/(3−x) (C) (x+1)/(3−2x) (D) (3x+1)/(x−2)

4. [Polynomial] The sum of all roots of x³−6x²+11x−6 is:

(A) 3 (B) 11 (C) 6 (D) 2

5. [Rational] For f(x) = (x²−4)/(x²−x−6), which best describes the graph?

(A) VAs at x=2, x=−3 (B) VA at x=3, hole at x=−2 (C) VAs at x=3, x=−2 (D) Hole at x=2, VA at x=−3

6. [Exponential] Solve 2^(2x−1) = 8^(x−2).

(A) x=3 (B) x=4 (C) x=5 (D) x=7

7. [Logarithm] Solve log₂x + log₂(x−6) = 4.

(A) x=8 (B) x=2 (C) x=8 or x=−2 (D) x=4

8. [Trig] Exact value of sin75°?

(A) (√3+1)/4 (B) (√6−√2)/4 (C) (√6+√2)/4 (D) (√2+√3)/2

9. [Trig Eq] How many solutions does 2sin²x − sinx − 1 = 0 have in [0, 2π)?

(A) 1 (B) 2 (C) 3 (D) 4

10. [Law of Cosines] In △ABC, a=7, b=5, C=60°. Find c².

(A) 74 (B) 39 (C) 49 (D) 29

11. [Arithmetic] a₁=3, d=4. Find S₁₀.

(A) 200 (B) 210 (C) 195 (D) 220

12. [Geometric] a₁=2, r=3. Find S₅.

(A) 240 (B) 242 (C) 244 (D) 246

13. [Binomial] Coefficient of x³ in (x+2)⁵?

(A) 10 (B) 20 (C) 40 (D) 80

14. [Polar] Convert (3, π/3) to rectangular coordinates.

(A) (3√3/2, 3/2) (B) (3/2, 3√3/2) (C) (3, 3√3/2) (D) (√3/2, 1/2)

15. [Parabola] For y²=12x, the focus is:

(A) (3, 0) (B) (0, 3) (C) (12, 0) (D) (0, 12)

16. [Ellipse] For x²/25 + y²/9 = 1, the foci are at:

(A) (±3, 0) (B) (±4, 0) (C) (0, ±4) (D) (±5, 0)

17. [Limits] lim(x→2) (x²−4)/(x−2) =

(A) undefined (B) 2 (C) 0 (D) 4

18. [Complex] Simplify (3+2i)(1−4i).

(A) 3−8i (B) 11+10i (C) 11−10i (D) −5−10i

19. [Matrices] det|2 3; 1 4| =

(A) 5 (B) 8 (C) 11 (D) −5

20. [Vectors] u = ⟨2,−1,3⟩, v = ⟨4,2,−1⟩. Find u·v.

(A) 3 (B) 7 (C) 13 (D) −3

Answer Key & Explanations

1. B — Need x≥3 (radicand) AND x≠5 (denom). Domain: [3,5)∪(5,∞).

2. B — g(2)=4−3=1; f(1)=3.

3. B — Swap x↔y, solve: f⁻¹(x) = (2x+1)/(3−x).

4. C — Vieta's: sum = −(−6)/1 = 6.

5. B — Factor to (x−2)/(x−3); (x+2) cancels → hole at x=−2, VA at x=3.

6. C — 8=2³; equate exponents: 2x−1=3x−6, so x=5.

7. A — x(x−6)=16 → x²−6x−16=0 → x=8 (x=−2 rejected, log undefined).

8. C — sin75°=sin(45°+30°)=(√6+√2)/4.

9. C — Factor: sinx=−½ gives x=7π/6,11π/6; sinx=1 gives x=π/2. Total: 3.

10. B — c²=49+25−2(7)(5)(½)=74−35=39.

11. B — a₁₀=39; S₁₀=5(3+39)=210.

12. B — S₅=2(3⁵−1)/(3−1)=242.

13. C — C(5,2)·2²=10·4=40.

14. B — x=3cos(π/3)=3/2; y=3sin(π/3)=3√3/2.

15. A — 4p=12 → p=3; focus=(3,0).

16. B — c²=25−9=16; c=4; foci=(±4,0).

17. D — Factor: (x−2)(x+2)/(x−2)=x+2 → limit=4.

18. C — FOIL: 3−12i+2i−8i²=11−10i.

19. A — det=2·4−3·1=5.

20. A — 8+(−2)+(−3)=3.

PrecalculusMaster Exam

Precalculus Master Exam — 20 Questions

Answer Key & Explanations

Precalculus
Master Exam