Function (Mathematics)

Context: FIT1058_MOC · the formal model of a computational task · three parts (domain, codomain, rule) · its graph is a subset of a Cartesian Product · a special Binary Relation

Quick Revision

• 🎯 Objective: assign each domain element exactly one codomain value ➔ models what a task does, not how.
• 📦 Core Components: domain ➔ codomain ➔ rule ➔ graph $\subseteq$ dom $\times$ codomain.
• ⚡ Key Constraint: totality + single-valuedness — every input has exactly one output.

📝 Core

1. The Function (Triple)

• Three parts ➔ domain (inputs), codomain (output space), rule ($x\mapsto f(x)$).
• Defining constraint ➔ every input has exactly one output (total + single-valued).
• What not how ➔ specifies input/output, no algorithm required.

2. The Graph (Set of Pairs)

• Definition ➔ $\text{graph}(f)=\{(x,f(x)):x\in \text{dom}\}\subseteq \text{dom}\times \text{codomain}$.
• Functional relation ➔ each domain element is the first coordinate of exactly one pair.
• Arrows ➔ exactly one arrow leaves each domain point (never 0, never 2).

3. Standard Functions

• Identity ➔ $i_A(x)=x$; constant ➔ $c_a(x)=a$.
• Indicator ➔ ${\chi }_A(x)=1$ if $x\in A$ else $0$; empty ➔ $\varnothing :\varnothing \to \varnothing$.

Key identities:

$f:\text{domain}\to \text{codomain},x\mapsto f(x);f:\mathbb{R}\to \mathbb{R},x\mapsto x^2$ $\text{graph}(f)=\{(x,f(x)):x\in \text{dom}(f)\}\subseteq \text{dom}(f)\times \text{codomain}$

⚖️ Core Decision Matrix

Aspect	Requirement	Consequence
totality	every input mapped	no missing outputs
single-valued	one output each	no ambiguity
domain part of identity	same rule, different domain	different function
multi-argument	domain is a Cartesian Product	$f(x,y)=f((x,y))$

When It Flips: a function is the triple (domain, codomain, rule), not the rule alone — same formula on different domains gives different functions. The rule ($x\mapsto x^2$) is the what; an algorithm is the how.

📊 Exam Execution Trace

Manual Execution Trace

$f:\{1,2,3\}\to \mathbb{Z}$, $f(x)=2x-1$:

Step / State	$x$	$f(x)$	Pair
0 (Init)	—	—	—
1	1	1	$(1,1)$
2	2	3	$(2,3)$
3	3	5	$(3,5)$

⚠️ Common Mistakes

• 💡 $\to$ links sets, $\mapsto$ links elements ➔ $A\to B$ is a signature; $x\mapsto f(x)$ maps one element to its value — not interchangeable.

🧠 Active Recall

Why are two functions with identical rules but different domains different functions?

• Hint: Function = triple.

Answer

• Short answer: The domain (and codomain) is part of a function’s identity, not just the rule.
• Why: Different graphs ➔ $w^3+\dots$ on ${\mathbb{Z}}^4$ vs ${\mathbb{R}}^4$ are different sets of pairs.

What is the graph, and what property makes a function more than an arbitrary set of pairs?

• Hint: Functional relation.

Answer

• Short answer: graph$=\{(x,f(x))\}$; each domain element is the first coordinate of exactly one pair.
• Why: Total + single-valued ➔ exactly one arrow leaves each domain point.

Distinguish a function's rule from an algorithm (squaring example).

• Hint: What vs how.

Answer

• Short answer: The rule $x\mapsto x^2$ fixes each output; an algorithm is a step-by-step method.
• Why: No method required ➔ many algorithms can realise the same rule.