# Function

The function is arguably the most fundamental concept in all of mathematics. Every mathematical structure — group, [topological](/page/Topology) space, measure space — is defined in terms of [sets](/page/Set) and functions. Yet the modern definition, which assigns to each element of a domain exactly one element of a codomain, took centuries to crystallise. Early mathematicians thought of functions as formulas: $y = x^2$, $y = \sin x$, expressions you could write down and compute. It was only through sustained engagement with [Fourier series](/page/Fourier%20Series), pathological constructions, and the foundations of set theory that the mathematical community arrived at the abstract, general definition we use today. This article develops that definition carefully, explores the key properties of injectivity, surjectivity, and bijectivity, and shows how the simple notion of a function ramifies into the structure-preserving maps that animate every branch of mathematics.

## Motivation

[motivation] ### From formulas to arbitrary assignments For most of the eighteenth century, a "function" meant a formula — an algebraic expression like $y = x^3 - 7x + 2$ or a trigonometric one like $y = \sin(x)$. Euler's original definition required that a function be given by a single analytic expression. This restriction was natural in an era when calculus was the study of curves, and curves were the graphs of formulas. The decisive break came with Fourier's work on heat conduction in 1807. Fourier claimed that an "arbitrary" function — even one defined piecewise, such as a function that equals $1$ on $[0, 1]$ and $0$ on $(1, 2]$ — could be represented by a trigonometric [series](/page/Series). His contemporaries found this deeply suspect, and the ensuing debate forced mathematicians to confront the question: what, exactly, is a function? Dirichlet's answer, given in 1837, was radical in its generality. A function from a set $X$ to a set $Y$, he proposed, is any rule that assigns to each element of $X$ a single element of $Y$. No formula is required. The assignment can be completely arbitrary — the only constraint is that each input produces exactly one output. This is the definition we formalise below. ### Why domain and codomain matter Consider the formula $f(x) = x^2$. Interpreted as a function $f \colon \mathbb{R} \to \mathbb{R}$, it is neither injective (since $f(-2) = f(2) = 4$) nor surjective (since $-1$ has no preimage). Interpreted as a function $f \colon [0, \infty) \to [0, \infty)$, it is both injective and surjective — a bijection. The formula has not changed, but the function has, because a function is not just a rule; it is a rule together with a specified domain and codomain. As we will see, injectivity, surjectivity, the existence of inverses, and even cardinality arguments all depend critically on the choice of domain and codomain. ### The power of abstraction Once we have the general definition of a function, powerful questions become available. When are two functions "the same"? The answer — when they have the same domain, the same codomain, and the same value at every point — is the principle of extensionality. When does a function have an inverse? Precisely when it is a bijection. And once we can compose functions and take inverses, we can ask which functions preserve mathematical structure. This last question leads to homomorphisms of groups, [continuous](/page/Continuity) maps of topological spaces, and [linear maps](/page/Linear%20Map) of vector spaces — the morphisms that form the backbone of modern mathematics. [/motivation]

## Formal Definition

[definition:Function] Let $X$ and $Y$ be sets. A **function** from $X$ to $Y$ is a subset $f \subseteq X \times Y$ satisfying two conditions: 1. **Existence.** For every $x \in X$, there exists a $y \in Y$ such that $(x, y) \in f$. 2. **Uniqueness.** If $(x, y_1) \in f$ and $(x, y_2) \in f$, then $y_1 = y_2$. The set $X$ is the **domain** of $f$, written $\operatorname{dom}(f) = X$. The set $Y$ is the **codomain** of $f$, written $\operatorname{cod}(f) = Y$. We write $f \colon X \to Y$ to indicate that $f$ is a function from $X$ to $Y$, and $f(x) = y$ to mean $(x, y) \in f$. The notation $x \mapsto f(x)$ describes the assignment rule. The **image** (or **range**) of $f$ is the set \begin{align*} \operatorname{im}(f) = \{y \in Y : \text{there exists } x \in X \text{ with } f(x) = y\}. \end{align*} [/definition]

The existence condition says that every element of the domain is "accounted for" — the function is defined everywhere on $X$. The uniqueness condition says that every input produces a single, well-defined output. Together, they ensure that the phrase "the value $f(x)$" makes sense for every $x \in X$.

It is important to distinguish the codomain from the image. The codomain $Y$ is part of the data of the function — it is declared, not computed. The image $\operatorname{im}(f)$ is the set of values actually attained, and it satisfies $\operatorname{im}(f) \subseteq Y$. The function $f \colon \mathbb{R} \to \mathbb{R}$ defined by $x \mapsto x^2$ has codomain $\mathbb{R}$ but image $[0, \infty)$. A function is surjective precisely when its image equals its codomain.

A word on notation: we write $f \colon X \to Y$ for the function itself, and $f(x)$ for the value of $f$ at a particular element $x \in X$. The function is the entire mapping $f$, not the expression $f(x)$. When we write "let $f \colon \mathbb{R} \to \mathbb{R}$ be the function $x \mapsto x^2$," the symbol $f$ names the function and $x \mapsto x^2$ describes its rule of assignment.

## Examples

[example:IdentityFunction] **The identity function.** For any set $X$, the **identity function** $\operatorname{id}_X \colon X \to X$ is defined by $\operatorname{id}_X(x) = x$ for all $x \in X$. As a subset of $X \times X$, it is the diagonal $\{(x, x) : x \in X\}$. The identity function is both injective and surjective, hence a bijection. It serves as the neutral element for function composition: $f \circ \operatorname{id}_X = f$ and $\operatorname{id}_Y \circ f = f$ for every function $f \colon X \to Y$. [/example]

[example:InclusionMap] **Inclusion maps.** Let $A \subseteq X$ be a subset. The **inclusion map** $\iota \colon A \hookrightarrow X$ is defined by $\iota(a) = a$ for all $a \in A$. This function is always injective: if $\iota(a_1) = \iota(a_2)$, then $a_1 = a_2$. It is surjective if and only if $A = X$. The inclusion map is not the same as the identity function unless $A = X$, because the two functions have different domains (and hence are different subsets of different Cartesian products). [/example]

[example:FloorFunction] **The floor function.** Define $\lfloor \cdot \rfloor \colon \mathbb{R} \to \mathbb{Z}$ by $\lfloor x \rfloor = \max\{n \in \mathbb{Z} : n \le x\}$. For instance, $\lfloor 3.7 \rfloor = 3$, $\lfloor -1.2 \rfloor = -2$, and $\lfloor 5 \rfloor = 5$. The floor function is surjective: every integer $n$ is the image of $n$ itself. It is not injective: both $3.1$ and $3.9$ map to $3$. [/example]

[example:ProjectionMap] **Projection maps.** Given sets $X$ and $Y$, the **projection maps** $\pi_1 \colon X \times Y \to X$ and $\pi_2 \colon X \times Y \to Y$ are defined by $\pi_1(x, y) = x$ and $\pi_2(x, y) = y$. When both $X$ and $Y$ are non-empty, each projection is surjective. However, neither is injective in general: if $|Y| \ge 2$, then the pairs $(x, y_1)$ and $(x, y_2)$ with $y_1 \ne y_2$ satisfy $\pi_1(x, y_1) = \pi_1(x, y_2) = x$. [/example]

## Injectivity, Surjectivity, and Bijectivity

The three most important properties a function can have concern how it relates the elements of its domain to the elements of its codomain. These properties control whether a function can be "reversed," and they play a central role in counting arguments, in the construction of inverse functions, and in the definition of cardinality.

[definition:InjectiveFunction] A function $f \colon X \to Y$ is **injective** (or **one-to-one**) if distinct elements of $X$ map to distinct elements of $Y$: \begin{align*} \text{for all } x_1, x_2 \in X, \quad f(x_1) = f(x_2) \implies x_1 = x_2. \end{align*} Equivalently, $f$ is injective if and only if the equation $f(x) = y$ has at most one solution $x \in X$ for every $y \in Y$. [/definition]

[definition:SurjectiveFunction] A function $f \colon X \to Y$ is **surjective** (or **onto**) if every element of $Y$ is hit: \begin{align*} \text{for all } y \in Y, \quad \text{there exists } x \in X \text{ such that } f(x) = y. \end{align*} Equivalently, $f$ is surjective if and only if $\operatorname{im}(f) = Y$. [/definition]

[definition:BijectiveFunction] A function $f \colon X \to Y$ is **bijective** (or a **bijection**, or a **one-to-one correspondence**) if it is both injective and surjective. Equivalently, $f$ is bijective if and only if for every $y \in Y$, the equation $f(x) = y$ has exactly one solution $x \in X$. [/definition]

To test injectivity in practice, one typically assumes $f(x_1) = f(x_2)$ and attempts to deduce $x_1 = x_2$. To test surjectivity, one takes an arbitrary $y \in Y$ and attempts to solve $f(x) = y$ for $x$.