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 $B$.

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 $x$ is a member of $X$, then the image of $x$ under $f$, denoted $f(x)$, is the value of $f$ when applied to $x$. $f(x)$ is alternatively known as the output of $f$ for argument $x$.

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

Image of a subset

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

$f[A]=\{f(x):x\in A\}$

When there is no risk of confusion, $f[A]$ is simply written as $f(A)$. This convention is a common one; the intended meaning must be inferred from the context. This makes $f[\,\cdot \,]$ a function whose domain is the power set of $X$ (the set of all subsets of $X$), and whose codomain is the power set 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 Y} .

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