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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\to\ Y} is a function from the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.