Geometric Bites

A Gentle Introduction to Function Notation Understanding f : A → B — the language of modern mathematics. One of the most powerful ideas in mathematics is the concept of a function . We usually meet it in the form f(x) = 2x + 1 , but the structure behind this idea is far richer. The notation f : A → B captures the entire architecture of a function in a single line. In this article, we unpack this notation and explain exactly what it means, why it matters, and how it connects to the familiar expression f(x) = y . 1. What does f : A → B mean? When we write f : A → B we are saying: f is a function, A is the domain — the set of inputs the function accepts, B is the codomain — the set in which all outputs must lie. In words: A function assigns to every element of the domain A exactly one output in the codomain B . Two rules always hold for genuine functions: Every input must have an output. No input may have more than one output. D...

Injective, Surjective, and Bijective Functions In mathematics, a function describes how elements of one set are assigned to elements of another. Three important properties capture how completely and uniquely a function connects its domain to its codomain: injective , surjective , and bijective . These properties tell us whether different inputs can share the same output and whether every possible output is used. Injective (One-to-One) A function is injective if different inputs always produce different outputs. No two distinct elements in the domain are allowed to map to the same result in the codomain. Formally, a function f : A → B is injective if: f(a₁) = f(a₂) ⇒ a₁ = a₂ Equivalently, if a₁ ≠ a₂ , then f(a₁) ≠ f(a₂) . Example: f(x) = e x from ℝ → ℝ is injective. Different inputs produce different outputs, but not every real number is reached, so it is not surjective. Surjective (Onto) A function is surjective if every element of the codomain is reach...