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 $x$ 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 $Y$ .

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

Examples

1. $f:\{1,2,3\}\to \{a,b,c,d\}$ defined by $f(x)=\left\{{\begin{matrix}a,&{\mbox{if }}x=1\\a,&{\mbox{if }}x=2\\c,&{\mbox{if }}x=3.\end{matrix}}\right.$

The image of the set

\{2,3\}

under

f

is

f(\{2,3\})=\{a,c\}.

The image of the function

f

is

\{a,c\}.

The preimage of

a

is

f^{-1}(\{a\})=\{1,2\}.

The preimage of

\{a,b\}

is also

f^{-1}(\{1,2\})=\{1,2\}.

The preimage of

\{b,d\},

is the empty set

\{\,\}=\varnothing .

2. $f:\mathbb {R} \to \mathbb {R}$ defined by $f(x)=x^{2}.$

The image of

\{-2,3\}

under

f

is

f^{-1}(\{-2,3\})=\{4,9\},

and the image of

f

is

\mathbb {R} ^{+}

(the set of all positive real numbers and zero). The preimage of

\{4,9\}

under

f

is

f^{-1}(\{4,9\})=\{-3,-2,2,3\}.

The preimage of set

N=\{n\in \mathbb {R} :n<0\}

under

f

is the empty set, because the negative numbers do not have square roots in the set of reals.

3. $f:\mathbb {R} ^{2}\to \mathbb {R}$ defined by $f(x,y)=x^{2}+y^{2}.$

The fiber

f^{-1}(\{a\})

are concentric circles]] about the origin, the origin itself, and the empty set, depending on whether

a>0,a=0,{\text{ or }}a<0,

respectively. (if

a>0,

then the fiber

f^{-1}(\{a\})

is the set of all

(x,y)\in \mathbb {R} ^{2}

satisfying the equation of the origin-concentric ring

x^{2}+y^{2}=a.

)

4. If $M$ is a manifold and $\pi :TM\to M$ is the canonical projection from the tangent bundle $TM$ to $M,$ then the fibers of $\pi$ are the tangent spaces $T_{x}(M){\text{ for }}x\in M.$ This is also an example of a fiber bundle.