Function¶

Definition¶

A function $f\colon X\to Y$ assigns to every input $x\in X$ exactly one output $f(x)\in Y\text{.}$ The phrase "exactly one" is essential: one input cannot be assigned two different outputs, although different inputs may share the same output.

Functions are also called maps. A function is specified by its domain, its codomain, and its assignment rule. Two functions $f,g\colon X\to Y$ with the same domain and codomain are equal, written $f=g\text{,}$ if $f(x)=g(x)$ for every $x\in X\text{;}$ a formula alone therefore does not determine a function.

Interpretation¶

A function is a rule that answers one question for each allowed input: which output belongs to it? Every input receives exactly one answer, so the output is completely determined by the input.

In the left diagram every point of $X$ has exactly one arrow leaving it. Two inputs share the middle output and the top point of $Y$ is never reached; both are allowed. In the right diagram the top input has two arrows and the middle input has none, so the rule fails in both ways and does not define a function from $X$ to $Y$.

The definition therefore makes two demands on every $x\in X$$X$: at least one output, so the rule must make sense on the whole domain, and at most one output. It demands nothing of the elements of $Y$. Whether every element of $Y$ is reached, and whether it is reached only once, are separate questions, answered by surjectivity and injectivity.

A common slip is to identify a function with its formula. The rule $x\mapsto x^2$ on $\mathbb{R}$ and the same rule on $[0,+\infty)$ are different functions: the second is injective, the first is not, since $(-1)^2=1^2$$1^2$. In the same way, $f$ denotes the function, while $f(x)$ is a single number, its value at $x$.

A sequence $(a_n)$ is itself a function: it assigns a term $a_n$ to each index $n$. The idea returns in later courses too, for example as linear maps between vector spaces in Math II.

Key Results¶

• Every function has a domain and a codomain.
• The range of a function is a subset of its codomain.

Examples¶

• $f(x) = x^2$ defined on the real numbers $\mathbb{R}\text{.}$
• $f(x) = \sqrt{x}$ defined on $[0, +\infty)\text{.}$

Prerequisites¶

• Real Number

Related Concepts¶

• Bijection and Inverse Function
• Sequence
• Binary Operation
• Cantor's Theorem
• Stochastic Process

Sources¶

• Math I: Calculus · Lecture 4

Aliases: map, mapping Keywords: transformation Symbols: element of, infinity, real numbers, square root

Last updated: 20 October 2026 03:02