Pre-Calculus Study Guide - 20 Essential Problems

01

Function Basics

Finding the Domain

Given: f(x) = √(x - 4)
Find the domain of this function.

KEY MEMORY: "Square root needs ≥ 0" → The expression inside must be ≥ 0, never negative!

Understanding the Problem:

Step 1: For square roots to be real, what's inside must be ≥ 0

Step 2: Set up the inequality: x - 4 ≥ 0

Step 3: Solve for x: x ≥ 4

Step 4: Write in interval notation: [4, ∞)

💡 Hint: Always check: fractions (denominator ≠ 0), square roots (inside ≥ 0), logarithms (argument > 0)

✏️ Your work space

02

Function Composition

Composing Two Functions

Given: f(x) = 2x + 1 and g(x) = x²
Find (f ∘ g)(x) and (g ∘ f)(x)

KEY MEMORY: "(f ∘ g)(x) = f(g(x))" → Plug g(x) into f, NOT the other way!

For (f ∘ g)(x):

Step 1: Write (f ∘ g)(x) = f(g(x))

Step 2: Substitute g(x) = x² into f(x) = 2x + 1

Step 3: f(x²) = 2(x²) + 1 = 2x² + 1

For (g ∘ f)(x):

Step 1: Write (g ∘ f)(x) = g(f(x))

Step 2: Substitute f(x) = 2x + 1 into g(x) = x²

Step 3: g(2x+1) = (2x+1)² = 4x² + 4x + 1

💡 Notice: (f ∘ g)(x) ≠ (g ∘ f)(x) → Order matters!

✏️ Your work space

03

Transformations

Vertical & Horizontal Shifts

Given: f(x) = x²
Describe transformations for: g(x) = (x-3)² + 2

KEY MEMORY: "(x - h) shifts RIGHT by h" + "Plus k shifts UP by k" (Opposite signs!)

Breaking Down the Transformation:

Step 1: Standard form: f(x - h) + k

Step 2: Here: (x - 3)² + 2 means h = 3 and k = 2

Step 3: Horizontal: (x-3) → shift RIGHT 3 units

Step 4: Vertical: +2 → shift UP 2 units

Step 5: Vertex moves from (0,0) to (3, 2)

Transformation Rules:

(x - h) → shift RIGHT h | (x + h) → shift LEFT h

+ k → shift UP k | - k → shift DOWN k

✏️ Your work space

04

Inverse Functions

Finding the Inverse Function

Given: f(x) = (3x - 2)/5
Find f⁻¹(x)

KEY MEMORY: "Swap x and y, then solve for y" → If y = f(x), then x = f(y) in inverse!

Step-by-Step Process:

Step 1: Replace f(x) with y: y = (3x - 2)/5

Step 2: Swap x and y: x = (3y - 2)/5

Step 3: Solve for y: 5x = 3y - 2

Step 4: 5x + 2 = 3y

Step 5: y = (5x + 2)/3, so f⁻¹(x) = (5x + 2)/3

💡 Check: If f(f⁻¹(x)) = x, you got it right!

✏️ Your work space

05

Polynomial Division

Synthetic Division Basics

Given: Divide x³ + 2x² - 5x + 3 by (x - 1)
Use synthetic division.

KEY MEMORY: "Synthetic division for (x - c) only" → Use the number c from (x - c), write coefficients in order!

Synthetic Division Setup:

Step 1: For (x - 1), use c = 1

Step 2: Write coefficients: 1, 2, -5, 3

Step 3: Bring down the 1

Step 4: Multiply, add, repeat: 1 × 1 = 1, then 2 + 1 = 3

Step 5: Continue: 3 × 1 = 3, then -5 + 3 = -2

Step 6: Last: -2 × 1 = -2, then 3 + (-2) = 1

Result: Quotient = x² + 3x - 2, Remainder = 1

✏️ Your work space

06

Rational Functions

Finding Vertical Asymptotes

Given: f(x) = (x + 2)/[(x-1)(x+3)]
Find all vertical asymptotes.

KEY MEMORY: "Vertical asymptote where denominator = 0" → Set denominator = 0, solve for x!

Finding Vertical Asymptotes:

Step 1: Vertical asymptotes occur where denominator = 0

Step 2: Set denominator: (x-1)(x+3) = 0

Step 3: Solve: x = 1 or x = -3

Step 4: Check numerator: At x = 1: 1 + 2 = 3 ≠ 0 ✓ (asymptote)

Step 5: At x = -3: -3 + 2 = -1 ≠ 0 ✓ (asymptote)

Answer: Vertical asymptotes: x = 1 and x = -3

💡 Important: If numerator is also 0 at that point, it's a HOLE, not an asymptote!

✏️ Your work space

07

Rational Functions

Finding Horizontal Asymptotes

Given: f(x) = (2x² + 3x)/(x² - 1)
Find the horizontal asymptote.

KEY MEMORY: "Compare degrees: equal → ratio of leading coefficients" → Look at highest powers!

Horizontal Asymptote Rule:

Step 1: Find degree of numerator: 2x² has degree 2

Step 2: Find degree of denominator: x² has degree 2

Step 3: Degrees are EQUAL

Step 4: Horizontal asymptote = ratio of leading coefficients = 2/1 = 2

Answer: y = 2

Three Rules for Horizontal Asymptotes:

• Numerator degree > Denominator → No horizontal asymptote

• Numerator degree = Denominator → y = (leading coeff num)/(leading coeff denom)

• Numerator degree < Denominator → y = 0

✏️ Your work space

08

Polynomial Zeros

Factoring and Finding Roots

Given: f(x) = x³ - 6x² + 11x - 6
Find all zeros (roots).

KEY MEMORY: "Rational Root Theorem: (factors of constant)/(factors of leading coeff)" → Test these values!

Using Rational Root Theorem:

Step 1: Possible rational roots: ±1, ±2, ±3, ±6

Step 2: Test x = 1: 1 - 6 + 11 - 6 = 0 ✓ (root found!)

Step 3: Factor out (x - 1): x³ - 6x² + 11x - 6 = (x-1)(x² - 5x + 6)

Step 4: Factor quadratic: x² - 5x + 6 = (x-2)(x-3)

Step 5: Complete factorization: (x-1)(x-2)(x-3)

Answer: Zeros are x = 1, 2, 3

✏️ Your work space

09

Exponential Equations

Solving Exponential Equations

Given: 2³ˣ = 32
Solve for x.

KEY MEMORY: "Make same base, then equate exponents" → If aᵐ = aⁿ, then m = n!

Step-by-Step Solution:

Step 1: Rewrite 32 as a power of 2: 32 = 2⁵

Step 2: Now equation is: 2³ˣ = 2⁵

Step 3: Bases are equal, so exponents must be equal

Step 4: 3x = 5

Step 5: x = 5/3

💡 Powers of 2: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, ...

✏️ Your work space

10

Logarithms

Converting Between Exponential & Logarithmic

Given: 3⁴ = 81
Write this in logarithmic form. Also convert: log₂ 8 = 3

KEY MEMORY: "aᵇ = c ↔ log_a c = b" → Base stays base, switch exponent and result!

Conversion Rules:

Exponential Form to Logarithmic:

If aᵇ = c, then log_a c = b

Example 1: 3⁴ = 81 becomes log₃ 81 = 4

Logarithmic Form to Exponential:

If log_a c = b, then aᵇ = c

Example 2: log₂ 8 = 3 becomes 2³ = 8

✏️ Your work space

11

Logarithmic Equations

Solving Logarithmic Equations

Given: log₃ (x + 2) = 2
Solve for x.

KEY MEMORY: "Convert to exponential form to solve" → log_a b = c means aᶜ = b!

Solution Process:

Step 1: Start with log₃ (x + 2) = 2

Step 2: Convert to exponential: 3² = x + 2

Step 3: Calculate: 9 = x + 2

Step 4: Solve: x = 7

Check: log₃(7+2) = log₃ 9 = log₃ 3² = 2 ✓

💡 Always check: The argument (what's inside log) must be > 0!

✏️ Your work space

12

Logarithm Properties

Expanding Logarithmic Expressions

Given: log_b [(x³y)/z²]
Expand using logarithm properties.

KEY MEMORY: "Log of product → add | Log of quotient → subtract | Log of power → multiply coefficient" → LOLM!

Three Key Properties:

1. Product Rule: log_b(MN) = log_b M + log_b N

2. Quotient Rule: log_b(M/N) = log_b M - log_b N

3. Power Rule: log_b(Mᵖ) = p log_b M

Applying to our problem:

Step 1: log_b [(x³y)/z²] = log_b(x³y) - log_b(z²) (Quotient Rule)

Step 2: = log_b(x³) + log_b(y) - log_b(z²) (Product Rule)

Step 3: = 3log_b(x) + log_b(y) - 2log_b(z) (Power Rule)

✏️ Your work space

13

Angle Conversion

Converting Degrees to Radians

Given: 45°
Convert to radians.

KEY MEMORY: "180° = π radians" → Multiply by π/180°!

Conversion Process:

Step 1: Use the conversion factor: (π radians)/(180°)

Step 2: Multiply: 45° × (π/180°)

Step 3: Simplify: (45π)/180 = π/4 radians

Key Conversions (Memorize!):

• 30° = π/6 | 45° = π/4 | 60° = π/3 | 90° = π/2 | 180° = π

✏️ Your work space

14

Trigonometric Ratios

Finding Trig Values from Right Triangles

Given: A right triangle with opposite side = 3, adjacent side = 4, hypotenuse = 5
Find sin θ, cos θ, tan θ

KEY MEMORY: "SOH-CAH-TOA" → Sin=Opposite/Hypotenuse, Cos=Adjacent/Hypotenuse, Tan=Opposite/Adjacent!

Using SOH-CAH-TOA:

Step 1: sin θ = (opposite)/(hypotenuse) = 3/5

Step 2: cos θ = (adjacent)/(hypotenuse) = 4/5

Step 3: tan θ = (opposite)/(adjacent) = 3/4

Reciprocal Functions:

• csc θ = 1/sin θ = 5/3

• sec θ = 1/cos θ = 5/4

• cot θ = 1/tan θ = 4/3

✏️ Your work space

15

Special Angles

Evaluating Trig Functions at Special Angles

Given: Find sin 60°, cos 45°, tan 30°

KEY MEMORY: "30-60-90 triangle: sides 1, √3, 2" and "45-45-90 triangle: sides 1, 1, √2" → Memorize special triangles!

Special Angle Triangles:

45-45-90 Triangle (Sides: 1, 1, √2):

sin 45° = cos 45° = 1/√2 = √2/2

tan 45° = 1

30-60-90 Triangle (Sides: 1, √3, 2):

sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3

sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Solutions:

Answer 1: sin 60° = √3/2

Answer 2: cos 45° = √2/2

Answer 3: tan 30° = √3/3 (or 1/√3)

✏️ Your work space

16

Trigonometric Identities

Using Pythagorean Identities

Given: sin² θ + cos² θ
Simplify this expression.

KEY MEMORY: "sin² θ + cos² θ = 1" → The fundamental identity! ALWAYS equals 1!

Pythagorean Identities:

Primary Identity:

sin² θ + cos² θ = 1

Derived Forms (Divide by cos² θ):

1 + tan² θ = sec² θ

Derived Forms (Divide by sin² θ):

1 + cot² θ = csc² θ

Example Application:

Step 1: If sin θ = 3/5, find cos θ

Step 2: Use sin² θ + cos² θ = 1

Step 3: (3/5)² + cos² θ = 1

Step 4: 9/25 + cos² θ = 1

Step 5: cos² θ = 16/25, so cos θ = 4/5 (assuming positive)

✏️ Your work space

17

Arithmetic Sequences

Finding Terms in Arithmetic Sequences

Given: An arithmetic sequence: 5, 9, 13, 17, ...
Find the 10th term and the general formula.

KEY MEMORY: "Arithmetic = constant difference d" → a_n = a_1 + (n-1)d ← USE THIS FORMULA!

Finding Terms:

Step 1: Find common difference: d = 9 - 5 = 4

Step 2: First term: a_1 = 5

Step 3: General formula: a_n = 5 + (n-1)(4) = 5 + 4n - 4 = 4n + 1

Step 4: 10th term: a_10 = 4(10) + 1 = 41

Verify: a_1 = 5, a_2 = 9, a_3 = 13 ✓

Arithmetic Sequence Formula:

a_n = a_1 + (n-1)d

where a_1 = first term, d = common difference, n = term number

✏️ Your work space

18

Geometric Sequences

Finding Terms in Geometric Sequences

Given: A geometric sequence: 2, 6, 18, 54, ...
Find the 7th term and the general formula.

KEY MEMORY: "Geometric = constant ratio r" → a_n = a_1 · rⁿ⁻¹ ← THIS FORMULA!

Finding Terms:

Step 1: Find common ratio: r = 6/2 = 3

Step 2: First term: a_1 = 2

Step 3: General formula: a_n = 2 · 3ⁿ⁻¹

Step 4: 7th term: a_7 = 2 · 3⁷⁻¹ = 2 · 3⁶ = 2 · 729 = 1458

Geometric Sequence Formula:

a_n = a_1 · rⁿ⁻¹

where a_1 = first term, r = common ratio, n = term number

✏️ Your work space

19

Series & Summation

Sum of Arithmetic Series

Given: Find the sum of the first 20 terms of: 3 + 7 + 11 + 15 + ...

KEY MEMORY: "Sum of arithmetic series = S_n = n(a_1 + a_n)/2" → Average first/last times count!

Solution Process:

Step 1: Identify: a_1 = 3, d = 4, n = 20

Step 2: Find a_20: a_20 = 3 + (20-1)(4) = 3 + 76 = 79

Step 3: Use sum formula: S_20 = 20(3 + 79)/2

Step 4: S_20 = (20)(82)/2 = 10(82) = 820

Sum of Arithmetic Series:

S_n = n(a_1 + a_n)/2 or S_n = (n/2)[2a_1 + (n-1)d]

✏️ Your work space

20

Binomial Theorem

Expanding Using Binomial Theorem

Given: Expand (x + 2)³

KEY MEMORY: "(a+b)ⁿ uses Pascal's Triangle for coefficients" → Or use C(n,k) formula!

Using Binomial Theorem:

Binomial Theorem:

(a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ (where k goes from 0 to n)

Step 1: For (x + 2)³: a = x, b = 2, n = 3

Step 2: Pascal's Triangle for n=3: coefficients are 1, 3, 3, 1

Step 3: Expand with decreasing powers of x, increasing powers of 2:

Step 4: = 1(x³)(2⁰) + 3(x²)(2¹) + 3(x¹)(2²) + 1(x⁰)(2³)

Step 5: = x³ + 6x² + 12x + 8

Pascal's Triangle (first few rows):

                            n=0: 1

                            n=1: 1 1

                            n=2: 1 2 1

                            n=3: 1 3 3 1

                            n=4: 1 4 6 4 1

✏️ Your work space