Functions:Forward Image

In mathematics, the image of a function is the set of all output values it may produce.

More generally, evaluating a given function $f$ at each element of a given subset $A$ of its domain produces a set, called the "image of $A$ under (or through) $f$". Similarly, the inverse image (or preimage) of a given subset $B$ of the codomain of $f$, is the set of all elements of the domain that map to the members of $B$.

Image and inverse image may also be defined for general binary relations, not just functions.

Definition

The word "image" is used in three related ways. In these definitions, $f:X\to \ Y$ is a function from the set $X$ to the set $Y$.

Image of an element If {\displaystyle x}x is a member of {\displaystyle X,}X, then the image of {\displaystyle x}x under {\displaystyle f,}f, denoted {\displaystyle f(x),}f(x), is the value of {\displaystyle f}f when applied to {\displaystyle x.}x. {\displaystyle f(x)}f(x) is alternatively known as the output of {\displaystyle f}f for argument {\displaystyle x.}x.

Given {\displaystyle y,}y, the function {\displaystyle f}f is said to "take the value {\displaystyle y}y" or "take {\displaystyle y}y as a value" if there exists some {\displaystyle x}x in the function's domain such that {\displaystyle f(x)=y.}{\displaystyle f(x)=y.} Similarly, given a set {\displaystyle S,}S, {\displaystyle f}f is said to "take a value in {\displaystyle S}S" if there exists some {\displaystyle x}x in the function's domain such that {\displaystyle f(x)\in S.}{\displaystyle f(x)\in S.} However, "{\displaystyle f}f takes [all] values in {\displaystyle S}S" and "{\displaystyle f}f is valued in {\displaystyle S}S" means that {\displaystyle f(x)\in S}{\displaystyle f(x)\in S} for every point {\displaystyle x}x in {\displaystyle f}f's domain.

Image of a subset The image of a subset {\displaystyle A\subseteq X}A\subseteq X under {\displaystyle f,}f, denoted {\displaystyle f[A],}{\displaystyle f[A],} is the subset of {\displaystyle Y}Y which can be defined using set-builder notation as follows:[1][2]

{\displaystyle f[A]=\{f(x):x\in A\}}{\displaystyle f[A]=\{f(x):x\in A\}} When there is no risk of confusion, {\displaystyle f[A]}{\displaystyle f[A]} is simply written as {\displaystyle f(A).}{\displaystyle f(A).} This convention is a common one; the intended meaning must be inferred from the context. This makes {\displaystyle f[\,\cdot \,]}{\displaystyle f[\,\cdot \,]} a function whose domain is the power set of {\displaystyle X}X (the set of all subsets of {\displaystyle X}X), and whose codomain is the power set of {\displaystyle Y.}Y. See § Notation below for more.

Image of a function The image of a function is the image of its entire domain, also known as the range of the function.[3] This usage should be avoided because the word "range" is also commonly used to mean the codomain of {\displaystyle f.}f.

Generalization to binary relations If {\displaystyle R}R is an arbitrary binary relation on {\displaystyle X\times Y,}{\displaystyle X\times Y,} then the set {\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}}{\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}} is called the image, or the range, of {\displaystyle R.}R. Dually, the set {\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}}{\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}} is called the domain of {\displaystyle R.}R.