function

function

A

function

is a

triplet

(

f

,

A

,

B

)

where:

1.

A

is a set (called the

domain

of the function).

2.

B

is a set (called the

codomain

of the function).

3.

f

is a

binary relation

between

A

and

B

.

4.

For every

a

∈

A

, there exists

b

∈

B

such that

(

a

,

b

)

∈

f

.

5.

If

a

∈

A

,

b

1

,

b

2

∈

B

, and

(

a

,

b

1

)

∈

f

and

(

a

,

b

2

)

∈

f

, then

b

1

=

b

2

.

The triplet

(

f

,

A

,

B

)

is usually written with the specialized notation

f

:

A

→

B

. This notation visually conveys the fact that

f

maps

elements of

A

into elements of

B

.

Other standard notations for functions are as follows:

•

For

a

∈

A

, one denotes by

f

⁢

(

a

)

the unique

element

b

∈

B

such that

(

a

,

b

)

∈

f

.

•

The

image

of

(

f

,

A

,

B

)

, denoted

f

⁢

(

A

)

, is the set

{

b

∈

B

∣

f

⁢

(

a

)

=

b

⁢

for some

⁢

a

∈

A

}

consisting of all elements of

B

which equal

f

⁢

(

a

)

for some element

a

∈

A

. Note that, by abuse of notation, the set

f

⁢

(

A

)

is almost always called the image of

f

, rather than the image of

(

f

,

A

,

B

)

.

•

In cases where the function

f

is clear from

context

, the notation

a

↦

b

is

equivalent

to the statement

f

⁢

(

a

)

=

b

.

•

Given two functions

f

:

A

→

B

and

g

:

B

→

C

, there exists a unique function

g

∘

f

:

A

→

C

satisfying the equation

g

∘

f

⁢

(

a

)

=

g

⁢

(

f

⁢

(

a

)

)

. The function

g

∘

f

is called the

composition

of

f

and

g

, and a function constructed in this manner is called a

composite function

. Composition is associative, meaning that

h

∘

(

g

∘

f

)

=

(

h

∘

g

)

∘

f

provided that either

expression

is defined.

•

When a function

f

:

A

→

A

has its domain equal to its codomain, one often writes

f

n

for the

n

-fold composition

f

∘

f

∘

⋯

∘

f

⏟

n

⁢

times

where

n

is any

natural number

. Occasionally this can be confused with ordinary

exponentiation

(for example the function

x

↦

(

sin

⁡

x

)

⁢

(

sin

⁡

x

)

is conventionally written as

sin

2

); in such cases one usually writes

f

[

n

]

to denote the

n

-fold composition.

There is no

universal

agreement as to the

definition

of the

range

of a function. Some authors define the range of a function to be equal to the codomain, and others define the range of a function to be equal to the image.

Remark

. In

set theory

, a function is defined as a

relation

f

, such that whenever

(

a

,

b

)

,

(

a

,

c

)

∈

f

, then

b

=

c

. Notice that the sets

A

,

B

are not specified in advance, unlike the defintion given in the beginning of the article. The

domain

and

range

of the function

f

is the domain and range of

f

as a relation. Using this definition of a function, we may recapture the defintion at the top of the entry by saying that a function

f

maps from a set

A

into a set

B

, if the domain of

f

is

A

, and the range of

f

is a subset of

B

.

Title

function

Canonical

name

Function

Date of creation

2013-03-22 11:48:58

Last modified on

2013-03-22 11:48:58

Owner

djao (24)

Last modified by

djao (24)

Numerical id

23

Author

djao (24)

Entry type

Definition

Classification

msc 03E20

Classification

msc 44A20

Classification

msc 33E20

Classification

msc 30D15

Synonym

map

Related topic

Mapping

Related topic

InjectiveFunction

Related topic

Surjective

Related topic

Bijection

Related topic

Relation

Defines

domain

Defines

codomain

Defines

composition

Defines

image

Defines

range

Defines

composite function

Generated on Thu Feb 8 20:12:54 2018 by

LaTeXML