Function Definition

Definition

A function $f$ from a set $A$ (domain) to a set $B$ is a rule that assigns a unique element $f(x) \in B$ to each element $x \in A$.

• How to read: “The function f maps each x in A to exactly one y in B.”
• Meaning: A function is a deterministic input-output rule: one input, one output. Example: area $A = \pi r^2$ depends on radius $r$.

Why It Matters

Functions are the fundamental unit of predictability in the universe; by formalizing the ‘one input, one output’ rule, they allow us to transition from a world of chaotic associations to deterministic models.

Core Concepts

• The Input-Output Rule: Every valid input $x$ (independent variable) produces a unique output $f(x)$ (dependent variable). The domain is all possible inputs; the range is all possible outputs.
  • How to read: “The output f of x is produced when x is the input.”
  • Meaning: $x$ is chosen freely (within domain); $f(x)$ is determined by the rule.
• Arrow Diagram: Connects an element $x \in A$ to its unique image $f(x) \in B$. Visual proof of single-valued property.
  • How to read: “Each element x in A points to its image f of x in B.”
  • Meaning: No input has two arrows leaving it.
• Vertical-Line Test: A geometric tool: if any vertical line intersects a graph at more than one point, the relation is not a function.

Connected Concepts

• Function Notation
• Function Domain
• Function Range
• One-to-One Functions
• Inverse Functions