f(x)
RULE ENGINE
The Function Machine
Relations, Functions & Mathematical Dependence
The Human Problem
World 08 taught us how to construct collections. Now we ask: What happens when elements of one collection are systematically related to elements of another? Mathematics creates a machine that takes an input and produces an output according to a defined rule. This gives us the language of dependence.
SET →
ORDERED PAIR →
RELATION →
FUNCTION →
MAPPING →
DOMAIN →
RANGE →
OPERATIONS →
COMPOSITION →
INVERSE →
TRANSFORMATION
The 51 Function Cells
From Ordered Pairs to The Function Universe
CELL 01
Ordered Pairs
(2, 5) ≠ (5, 2)
- (a, b) notation
- Order determines meaning
- Foundation of relations
CELL 02
Cartesian Product
- A × B
- Universe of possible pairings
- Source set for relations
CELL 03
Relations
- Subset of A × B
- Rule connecting elements
- Set of ordered pairs
CELL 04
Domain
Set A (Inputs){1, 2, 3}
Set B (Outputs){a, b, c}
- Set of first components
- Valid input universe
CELL 05
Range
Set A (Inputs){1, 2, 3}
Set B (Outputs){a, b, c}
- Set of actual outputs
- Second components
CELL 06
Codomain
Range{2, 4}
Codomain{2, 4, 6, 8}
- Declared target universe
- May be larger than Range
CELL 07
Function
- Special relation
- Every input has EXACTLY ONE output
- Predictable machine
CELL 08
Mapping Diagram
123
AB
- Visualizing input to output
- Detecting function validity
CELL 09
Function Notation
f(x)
- f(x) replaces y
- Names the rule
- Evaluation e.g., f(4)
CELL 11
Domain Restrictions
1x - 2
x ≠ 2
- Avoiding division by zero
- Avoiding negative even roots
- Defining valid input set
CELL 13
Range vs Codomain
RangeOutputs reached
CodomainTarget set
Range ⊆ Codomain
- Crucial formal distinction
- Determines 'onto' property
CELL 14
Representations
Formula
Table
Graph
Map
- Equation (y = 2x)
- Set of ordered pairs
- Cartesian graph
CELL 15
Vertical Line Test
- Visual function check
- Must cross curve max once
- Circle fails test
CELL 16
Constant Function
- f(x) = c
- Horizontal line graph
- All inputs map to one output
CELL 17
Identity Function
- f(x) = x
- Input equals output
- Neutral element in composition
CELL 18
One-One Function
- Injective
- Distinct inputs → distinct outputs
- Horizontal line test passes
CELL 19
Many-One Function
- Multiple inputs → same output
- Example: f(x) = x²
- Not invertible globally
CELL 20
Onto Function
- Surjective
- Range = Codomain
- Every possible output is reached
CELL 22
Bijective Function
- One-One AND Onto
- Perfect 1-to-1 pairing
- Condition for invertibility
CELL 23
Reflexive Relation
- aRa for all a ∈ A
- Every element relates to itself
CELL 24
Symmetric Relation
- If aRb, then bRa
- Two-way connection
CELL 25
Transitive Relation
- If aRb and bRc, then aRc
- Bridging connections
CELL 26
Equivalence Relation
REFLEXIVE + SYMMETRIC + TRANSITIVE
- Creates equivalence classes
- Treating different objects as 'same'
CELL 28
Composition
x
g
g(x)f
f(g(x))
- (f ∘ g)(x) = f(g(x))
- Chaining function machines
- Generally not commutative
CELL 31
Inverse Functions
- f⁻¹(x)
- Reverses the action of f
- Requires function to be bijective
CELL 34
Graph of Inverse
- Reflection across line y = x
- Domain & Range swap
CELL 36
Restricting Domain
- Making non-injective functions invertible
- e.g., restricting x² to x ≥ 0
CELL 37
Piecewise Functions
- Different rules for different intervals
- Foundation for continuity checks
CELL 38
Modulus Function
- f(x) = |x|
- V-shaped graph
- Distance from zero
CELL 39
Polynomial Functions
- Smooth, continuous curves
- Degree determines shape
CELL 41
Exponential Functions
- f(x) = a^x
- Rapid growth / decay
- Inverse of logarithmic function
CELL 44
Graph Transformations
- Shifting: f(x) + k, f(x-h)
- Scaling: a f(x), f(bx)
- Reflecting: -f(x), f(-x)
CELL 46
Periodic Functions
- f(x+T) = f(x)
- Repeating cycles (Sine, Cosine)
- Applications in physics
CELL 50
Composition as Systems
T(x)→
ENERGY
→
COST
→
C(E(T(x)))
- Modeling real-world chains
- Output becomes next input
- Foundation of systems thinking
Exit Portal
We now possess the language of Dependence. We can describe how one quantity changes precisely as another changes. Armed with functions, we are ready to leave pure algebra and enter the space where numbers describe shapes, structures, and dimensions.