Codomain¶

Definition¶

If the function is $f\colon X\to Y\text{,}$ the codomain of $f$ is the specified set $Y$ in which every output must lie. The range is always a subset of the codomain, but the two sets need not be equal.

Interpretation¶

The codomain is the set that a function declares as the place where its outputs live. It is part of the specification $f\colon X\to Y$$Y$, together with the domain and the rule, and is usually chosen as a large, convenient set such as $\mathbb{R}$.

In the figure, the three points of $X$ are sent into $Y$, but only three points of $Y$ are hit. They form the range $f(X)$, the smaller box inside $Y$; the grey points belong to the codomain but are not values of $f$.

The codomain is chosen, while the range follows from the rule and has to be worked out. A function is surjective exactly when its range is the whole codomain, so changing the codomain can change surjectivity without changing any value: $x\mapsto x^2$ is not surjective as a map from $\mathbb{R}$ to $\mathbb{R}$, since $-1$ is never reached, but it is surjective onto $[0,\infty)$. This matters for a bijection, whose inverse must be defined on the whole codomain.

A common slip is to call the codomain the set of values of $f$, which is the range. The sine function from $\mathbb{R}$ to $\mathbb{R}$ has codomain $\mathbb{R}$, but its range is $[-1,1]$.

Key Results¶

• The range of a function is always a subset of its codomain.
• Changing the codomain can affect whether a function is surjective.

Examples¶

• For $f(x)=x^2$ defined as $f\colon \mathbb{R}\to\mathbb{R}\text{,}$ the codomain is $\mathbb{R}\text{.}$
• Strictly speaking, the same formula with codomain $[0,+\infty)$ defines a different function.

Prerequisites¶

• Function

Related Concepts¶

• Domain
• Range and Image
• Random Variable

Sources¶

• Math I: Calculus · Lecture 4

Aliases: target set Symbols: infinity, real numbers

Last updated: 20 October 2026 03:02