1. Real Numbers & The Coordinate Plane

Real Number System Hierarchy

Real Numbers (ℝ): All numbers on the number line, including both rational and irrational numbers.

Number Set	Symbol	Description	Examples
Natural Numbers	ℕ	Counting numbers starting at 1	1, 2, 3, 4, 5, ...
Whole Numbers	W	Natural numbers plus zero	0, 1, 2, 3, 4, ...
Integers	ℤ	Whole numbers and their opposites	..., -2, -1, 0, 1, 2, ...
Rational Numbers	ℚ	Numbers that can be written as a/b where b ≠ 0	1/2, -3, 0.75, 2/3
Irrational Numbers	—	Cannot be written as a fraction; non-repeating, non-terminating decimals	√2, π, √7, e

Hierarchy: ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ
(Each set contains all sets to its left)

The Cartesian Coordinate Plane

Ordered Pair: (x, y) where x is the horizontal coordinate and y is the vertical coordinate.

Four Quadrants

Quadrant I: (+, +) - both coordinates positive
Quadrant II: (−, +) - x negative, y positive
Quadrant III: (−, −) - both coordinates negative
Quadrant IV: (+, −) - x positive, y negative

Points on axes are NOT in any quadrant:
• x-axis: y = 0
• y-axis: x = 0

Essential Formulas

Distance Formula

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example: Distance between (1, 2) and (4, 6)
d = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5

Midpoint Formula

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Example: Midpoint of (2, 3) and (8, 7)
M = ((2+8)/2, (3+7)/2) = (10/2, 10/2) = (5, 5)

2. Relations & Introduction to Functions

Definitions

Relation: A set of ordered pairs (x, y). Can be represented as a list, table, graph, or mapping.

Function: A special relation where each input (x-value) corresponds to EXACTLY ONE output (y-value).

KEY RULE: In a function, no x-value can repeat with different y-values!

IS a function: {(1,2), (2,3), (3,4)}
NOT a function: {(1,2), (1,3), (2,4)} (x=1 appears twice)

Vertical Line Test

Vertical Line Test: A graph represents a function if and only if NO vertical line intersects the graph more than once.

Graph Type	Is it a Function?
Parabola opening up/down (y = x²)	YES
Circle (x² + y² = r²)	NO
Straight line (not vertical)	YES
Vertical line (x = 3)	NO
Sideways parabola (x = y²)	NO

Function Notation

f(x): Read as "f of x" - represents the output value when x is the input.
f(x) replaces y in equations.

Example: If f(x) = 2x + 3, then:
• f(5) = 2(5) + 3 = 13
• f(-1) = 2(-1) + 3 = 1
• f(a) = 2a + 3

Domain and Range

Domain: The set of all possible INPUT values (x-values) for a function.

Range: The set of all possible OUTPUT values (y-values) for a function.

Common Domain Restrictions

Division by zero: For f(x) = 1/(x-3), domain excludes x = 3
Square roots: For f(x) = √(x-2), domain is x ≥ 2 (expression under √ must be ≥ 0)
No restrictions: Polynomials like f(x) = x² + 3x - 1 have domain = all real numbers

Examples:
• f(x) = x² → Domain: ℝ, Range: [0, ∞)
• g(x) = 1/x → Domain: x ≠ 0, Range: y ≠ 0
• h(x) = √x → Domain: [0, ∞), Range: [0, ∞)

3. Function Arithmetic

Four Operations on Functions

1. (f + g)(x) = f(x) + g(x) [Addition]
2. (f - g)(x) = f(x) - g(x) [Subtraction]
3. (f · g)(x) = f(x) · g(x) [Multiplication]
4. (f / g)(x) = f(x) / g(x), where g(x) ≠ 0 [Division]

Example: If f(x) = x² and g(x) = 2x + 1
• (f + g)(x) = x² + 2x + 1
• (f - g)(x) = x² - 2x - 1
• (f · g)(x) = x²(2x + 1) = 2x³ + x²
• (f / g)(x) = x²/(2x + 1), where x ≠ -1/2

Domain of Combined Functions

For addition, subtraction, and multiplication:
Domain = intersection of individual domains

For division (f/g):
Domain = intersection of individual domains, EXCLUDING values where g(x) = 0

Example: f(x) = √x (domain: [0, ∞)) and g(x) = x - 4 (domain: all reals)
• (f + g)(x) domain: [0, ∞)
• (f / g)(x) domain: [0, ∞) excluding x = 4, so [0, 4) ∪ (4, ∞)

4. Graphs of Functions & Transformations

Six Parent Functions

Name	Equation	Shape/Description
Linear	f(x) = x	Diagonal line through origin, slope 1
Quadratic	f(x) = x²	U-shaped parabola, vertex at origin
Cubic	f(x) = x³	S-curve through origin
Absolute Value	f(x) = \|x\|	V-shape, vertex at origin
Square Root	f(x) = √x	Half-parabola starting at origin, going right
Reciprocal	f(x) = 1/x	Hyperbola in Quadrants I and III

Transformation Rules

Vertical Shifts

f(x) + k

k > 0: Shift UP k units
k < 0: Shift DOWN k units

Example: f(x) = x² + 3 shifts the parabola UP 3 units

Horizontal Shifts

f(x - h)

h > 0: Shift RIGHT h units (OPPOSITE of sign!)
h < 0: Shift LEFT h units

Example: f(x) = (x - 2)² shifts the parabola RIGHT 2 units
Tricky: f(x) = (x + 3)² shifts LEFT 3 units (not right!)

HORIZONTAL SHIFTS ARE OPPOSITE OF THE SIGN!
f(x - h) → shift RIGHT
f(x + h) → shift LEFT

Vertical Stretch/Compression

a · f(x)

|a| > 1: Vertical STRETCH (taller, narrower)
0 < |a| < 1: Vertical COMPRESSION (shorter, wider)

Example: g(x) = 3x² stretches the parabola vertically by factor of 3

Reflections

Transformation	Effect
-f(x)	Reflect over x-axis (flip upside down)
f(-x)	Reflect over y-axis (flip left-to-right)

Example: g(x) = -x² makes the parabola open DOWNWARD

Order of Transformations

CRITICAL - Apply in this order:
1. Horizontal shifts (inside parentheses)
2. Stretches/compressions and reflections (coefficient)
3. Vertical shifts (outside addition/subtraction)

Example: g(x) = -2(x - 3)² + 1 starting from f(x) = x²

Step 1: Shift RIGHT 3 → (x - 3)²
Step 2: Stretch by 2 → 2(x - 3)²
Step 3: Reflect over x-axis → -2(x - 3)²
Step 4: Shift UP 1 → -2(x - 3)² + 1

Result: Upside-down parabola with vertex at (3, 1)

Common Mistakes to Avoid

Forgetting that horizontal shifts are OPPOSITE of the sign
Confusing f(x) + k (vertical shift) with f(x + k) (horizontal shift)
Mixing up -f(x) (reflection over x-axis) with f(-x) (reflection over y-axis)
Applying transformations in the wrong order
Forgetting domain restrictions when dividing functions

Practice Problems with Solutions

Problem 1: Classify the number √3

Answer: Irrational and Real

Explanation: √3 cannot be written as a fraction (approximately 1.732...), so it's irrational. All irrational numbers are real.

Problem 2: Is {(1,2), (2,2), (3,2)} a function?

Answer: YES

Explanation: Each x-value (1, 2, 3) appears only once. Multiple inputs can map to the same output (all map to 2).

Problem 3: Find the domain of f(x) = 1/(x² - 9)

Answer: All real numbers except x = 3 and x = -3

Explanation: Set denominator ≠ 0: x² - 9 ≠ 0, so x² ≠ 9, meaning x ≠ ±3

Problem 4: Describe the transformation of g(x) = (x + 2)² - 5 from f(x) = x²

Answer: Shift LEFT 2 units and DOWN 5 units

Explanation: (x + 2) means left 2 (opposite of sign), -5 outside means down 5. Vertex moves from (0,0) to (-2,-5).

Module 1: Introduction to Functions & The Coordinate Plane

Free College Algebra Learning Platform • Safaa Dabagh • sdabagh.github.io

Module 1 Study Guide