Témbongkeun sumber pikeun Fungsi (matematika)

Dina [[matematik]], '''fungsi''' nyaéta hiji ''hubungan,'' saperti unggal [[unsur (matematik)|unsur]] tina [[set|susunan]] mangrupa gabungan tina susunan unsur unik séjénna (nu mungkin sarua). Konsép fungsi jadi dasar pikeun sababaraha widang matematik sarta sakabéh [[élmu]] kuantitatif.

Watesan '''fungsi''', '''mapping''', '''map''', '''transformation''' sarta '''operator''' mangrupa hal nu ilaharna meh sarua.

== Intuitive introduction ==
Essentially, a function is a "rule" that assigns a [[unique]] [[output]] to éach given [[input]]. Here are some examples of functions:

* éach person has a favorite colour (red, orange, yellow, green, cyan, blue, indigo, or violet). The colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Here, the input is the person, and the output is one of the 8 colours.
* Some children are selling lemonade in the summer. The number of lemonades they sell is a function of the temperature outside. For example, if it is 85 degrees outside, they sell 10 lemonades, but if it is 95 degrees outside, they sell 25 lemonades. Here, the input is the temperature, and the output is the number of lemonades they sell.
* A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second storey, and (only) 4 seconds to fall from the 10th storey. Here, the input is the storey, and the output is the number of seconds. The relevant ''function'' describes the relationship between the time it takes the stone to réach the ground and the storey. (''See [[acceleration]]'')

The "rule" defining a function can be specified by a [[formula]], a [[mathematical relation|relationship]], or simply a table listing the outputs against inputs. The most important féature of a function is that it is [[determinism|deterministic]], always producing the same [[output]] from the same [[input]]. In this way, a function may be thought of as a "[[machine]]" or a "[[black box]]", converting a valid input into a unique output. The input is often called the ''argument'' of the function, and the output the ''value'' of the function.

A very common type of function occurs when the argument and the function value are both [[number]]s, the functional relationship is expressed by a formula, and the value of the function is obtained by direct substitution of the argument into the formula. Consider for example
:<math>f(x)=x^{2}</math>
which assigns to any number ''x'' its square.

A straightforward generalization is to allow functions depending on several arguments. For instance,
:<math>g(x,y) = xy</math> 
is a function which takes two numbers ''x'' and ''y'' and assigns to them their product, ''xy''. It might seem that this is not réally a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single ''[[ordered pair|pair]]'' (''x'', ''y''), then we can interpret ''g'' as a function—the argument is the ordered pair (''x'', ''y''), and the function value is ''xy''.

In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on éarth over time: this is a function which takes location and time as arguments and gives as output the temperature at that location at that time.

We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every [[naive set theory|element]] of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain [[mathematical relation|relations]], as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics.

== Sajarah ==

As a mathematical term, "'''function'''" was coined by [[Gottfried Leibniz|Leibniz]] in [[1694]], to describe a quantity related to a [[curve]], such as a curve's [[slope]] or a specific [[point]] of a curve. The functions Leibniz considered are today called [[derivative|differentiable functions]], and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about [[limit of a function|limits]] and [[derivative]]s; both are méasurements of the change of output values associated to a change of input values, and these méasurements are the basis of [[calculus]].

The word function was later used by [[Leonhard Euler|Euler]] during the mid-[[18th century]] to describe an [[Expression (mathematics)|expression]] or formula involving various [[parameter|arguments]], e.g. ''f''(''x'') = sin(''x'') + ''x''<sup>3</sup>.

During the [[19th century]], mathematicians started to formalize all the different branches of mathematics. [[Karl Weierstrass|Weierstrass]] advocated building calculus on [[arithmetic]] rather than on [[geometry]], which favoured Euler's definition over Leibniz's (see [[arithmetization of analysis]]).

By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only théoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from [[functional analysis]] has shown that these functions are actually more common than differentiable functions. Such functions have since been applied to the modélling of physical phenomena such as [[Brownian motion]].

Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using [[axiomatic set theory|set theory]], and they sought to define every mathematical object as a [[set]]. [[Johann Peter Gustav Lejeune Dirichlet|Dirichlet]] and [[Nikolai Ivanovich Lobachevsky|Lobachevcky]] independently and almost simultanéously gave the modérn "formal" definition of function (see [[#Formal definition|formal definition]] below).

In this definition, a function is a special case of a [[mathematical relation|relation]]. In most cases of practical interest, however, the differences between the modérn definition and Euler's definition are negligible.

The notion of '''function''' as a rule for computing, rather than a special kind of relation, has been formalized in [[mathematical logic]] by méans of the [[lambda calculus]].

== Dadaran formal ==

Formally, a function ''f'' from a set ''X'' of input values to a set ''Y'' of possible output values (written as ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''Y'') is a [[binary relation|relation]] between ''X'' and ''Y'' which satisfies:
# ''f'' is ''total'': for all ''x'' in ''X'', there exists a ''y'' in ''Y'' such that ''x f y'' (''x'' is ''f''-related to ''y''), i.e. for éach input value, there is at léast one output value in ''Y''.
# ''f'' is ''many-to-one'': if ''x f y'' and ''x f z'', then ''y'' = ''z''. i.e., many input values can be related to one output value, but one input value cannot be related to many output values.

For éach input value ''x'' in the domain, the corresponding unique output value ''y'' in the codomain is denoted by ''f''(''x'').

A more concise expression of the above definition is the following: a function from ''X'' to ''Y'' is a [[subset]] ''f'' of the [[cartesian product]] ''X'' × ''Y'', such that for éach ''x'' in ''X'', there is a unique ''y'' in ''Y'' such that the ordered pair (''x'', ''y'') is in ''f''.

The set of all functions ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''Y'' is denoted by ''Y<sup>X</sup>''. Note that ''|Y<sup>X</sup>|&nbsp;=&nbsp;|Y|<sup>|X|</sup>'' (refer to [[Cardinal number#Formal definition|Cardinal numbers]]).

A relation between ''X'' and ''Y'' that satisfies condition (1) is a '''[[multivalued function]]'''. Every function is a multivalued function, but not every multivalued function is a function. A relation between ''X'' and ''Y'' that satisfies condition (2) is a '''[[partial function]]'''. Every function is a partial function, but not every partial function is a function. In this encyclopedia, the term "function" will méan a relation satisfying both conditions (1) and (2), unless otherwise stated.

Consider the following three examples:

<table>

<tr><td>[[Gambar:notMap1.png]]</td><td> This relation is total but not many-to-one; the element 3 in ''X'' is related to two elements ''b'' and ''c'' in ''Y''. Therefore, this is a multivalued function, but ''not'' a function.</td></tr>

<tr><td>[[Gambar:notMap2.png]]</td><td> This relation is many-to-one but not total; the element 1 in ''X'' is not related to any element of ''Y''. Therefore, this is a partial function, but ''not'' a function.<td></tr>

<tr><td>[[Gambar:Injection.svg]]</td><td> This relation is both total and many-to-one, and so it is a function from ''X'' to ''Y''. The function can be given explicitly as ''f'' = {(1, a), (2, d), (3, c)} or as
:<math>f(x)=\left\{\begin{matrix} a, & \mbox{if }x=1 \\ d, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \end{matrix}\right.</math>
</td></tr>
</table>

== Domains, codomains, and ranges ==

''X'', the set of input values, is called the [[Function domain|domain]] of ''f'', and ''Y'', the set of '''possible''' output values, is called the [[codomain]]. The [[function range|range]] of ''f'' is the set of all '''actual''' outputs {''f''(''x'') : ''x'' in the domain}. Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. 
An [[endofunction]] is a function whose domain and range are identical.

In computer science, the [[datatype]]s of the arguments and return values specify the domain and codomain (respectively) of a [[subprogram]]. So the domain and codomain are constraints imposed initially on a function; on the other hand the range has to do with how things turn out in practice.

== Injective, surjective and bijective functions ==

Several types of functions that are very useful have special names: 
* [[Injective]] (one-to-one) functions send different arguments to different values; in other words, if ''x'' and ''y'' are members of the domain of ''f'', then ''f''(''x'') = ''f''(''y'') only if ''x'' = ''y''.
* [[Surjective]] (onto) functions have their range equal to their codomain; in other words, if ''y'' is any member of the codomain of ''f'', then there exists at léast one ''x'' such that ''f''(''x'') = ''y''.
* [[Bijective function]]s are both injective and surjective; they are often used to show that the sets ''X'' and ''Y'' are the "same size" in some sense.

== Images and preimages ==

The ''[[image (mathematics)|image]]'' of an element ''x''∈''X'' under ''f'' is the output ''f''(''x'').

The image of a subset <var>A</var>⊂''X'' under ''f'' is the subset of ''Y'' defined by
:''f''(<var>A</var>)&nbsp;:= {''f''(<var>x</var>)&nbsp;: <var>x</var> in <var>A</var>}. 
 
Notice that the range of ''f'' is the image ''f''(''X'') of its domain. In our function above, the image of {2,3} under ''f'' is ''f''({2, 3}) = {c, d} and the range of ''f'' is {a, c, d}.

Note that with this definiton, the direct image ''f'' becomes a function whose domain is the set of all subsets of ''X'' (also known as the [[power set]] of ''X'') and whose codomain is the power set of ''Y''. Note that the same notation is used for the original function ''f'' and its direct image. This is a common convention; the intended usage must be inferred by context.

The ''preimage'' (or ''inverse image'') of a set <var>B</var> ⊂ <var>Y</var> under ''f'' is the subset of <var>X</var> defined by
:''f''<sup>&nbsp;−1</sup>(<var>B</var>)&nbsp;:= {<var>x</var> in ''X''&nbsp;: ''f''(<var>x</var>)∈<var>B</var>}.
In our function above, the preimage of {a, b} is ''f''<sup>&nbsp;−1</sup>({a, b}) = {1}.

Note that with this definiton, ''f''<sup>&nbsp;−1</sup> becomes a function whose domain is the power set of ''Y'' and whose codomain is the power set of ''X'''.

Some consequences that follow immediately from these definitions are:
* <var>f</var>(<var>A</var><sub>1</sub>&nbsp;∪&nbsp;<var>A</var><sub>2</sub>)&nbsp;= <var>f</var>(<var>A</var><sub>1</sub>)&nbsp;∪&nbsp;<var>f</var>(<var>A</var><sub>2</sub>).
* <var>f</var>(<var>A</var><sub>1</sub>&nbsp;∩&nbsp;<var>A</var><sub>2</sub>)&nbsp;⊆ <var>f</var>(<var>A</var><sub>1</sub>)&nbsp;∩&nbsp;<var>f</var>(<var>A</var><sub>2</sub>).
* <var>f</var><sup>&nbsp;−1</sup>(<var>B</var><sub>1</sub>&nbsp;∪&nbsp;<var>B</var><sub>2</sub>)&nbsp;= <var>f</var><sup>&nbsp;−1</sup>(<var>B</var><sub>1</sub>)&nbsp;∪&nbsp;<var>f</var><sup>&nbsp;−1</sup>(<var>B</var><sub>2</sub>).
* <var>f</var><sup>&nbsp;−1</sup>(<var>B</var><sub>1</sub>&nbsp;∩&nbsp;<var>B</var><sub>2</sub>)&nbsp;= <var>f</var><sup>&nbsp;−1</sup>(<var>B</var><sub>1</sub>)&nbsp;∩&nbsp;<var>f</var><sup>&nbsp;−1</sup>(<var>B</var><sub>2</sub>).
* <var>f</var>(<var>f</var><sup>&nbsp;−1</sup>(<var>B</var>))&nbsp;⊆&nbsp;<var>B</var>.
* <var>f</var><sup>&nbsp;−1</sup>(<var>f</var>(<var>A</var>))&nbsp;⊇&nbsp;<var>A</var>.

These are valid for arbitrary subsets ''A'', ''A''<sub>1</sub> and ''A''<sub>2</sub> of the domain and arbitrary subsets ''B'', ''B''<sub>1</sub> and ''B''<sub>2</sub> of the codomain.
The results relating images and preimages to the algebra of [[set theoretic intersection|intersection]] and [[set theoretic union|union]] work for any collections of subsets, not just for pairs of subsets.

== Graph of a function ==

The [[graph of a function]] ''f'' is the set of all [[ordered pair]]s(''x'', ''f''(''x'')), for all ''x'' in the domain ''X''. There are théorems formulated or proved most éasily in terms of the graph, such as the [[closed graph theorem]].

If ''X'' and ''Y'' are réal lines, then this definition coincides with the familiar sense of graph. Below is the graph of a cubic function:

[[Gambar:cubicpoly.png]]

This function is surjective but not injective.

Note that since a relation on the two sets ''X'' and ''Y'' is usually formalized as a subset of ''X''×''Y'', the formal definition of function actually identifies the function ''f'' with its graph.

== Conto fungsi ==

(More can be found at [[List of functions]].)

* The relation ''wght'' between persons in the United States and their weights at a particular time.
* The relation between nations and their capitals, if we exclude those nations that maintain multiple capitals [http://geography.about.com/library/misc/bl2capitals.htm].
* The relation ''sqr'' between [[natural number]]s <var>n</var> and their squares <var>n</var><sup>2</sup>.
* The relation ''ln'' between ''positive'' [[real number]]s <var>x</var> and their [[natural logarithm]]s ln(<var>x</var>). Note that the relation between réal numbers and their natural logarithms is not a function because not every réal number has a natural logarithm; that is, this relation is not total.
* The relation ''dist'' between points in the plane '''R'''<sup>2</sup> and their distances from the origin (0,0).
* The relation ''grav'' between a point in the punctured plane '''R'''<sup>2</sup>&nbsp;\ {(0,0)} and the vector describing the [[gravitational force]] that a certain mass at that point would experience from a certain other mass at the origin (0,0).

Most commonly used types of mathematical functions involving [[addition]], [[division]], [[exponent]]s, [[logarithm]]s, [[multiplication]], [[polynomial]]s, [[radical]]s, [[rational]]s, [[subtraction]], and [[trigonometric functions|trigonometric expressions]]. They are sometimes collectively referred as '''elementary functions''' -- but the méaning of this term varies among different branches of mathematics. Example of non-elementary functions (or [[special function]]s) are [[Bessel function]]s sarta [[fungsi gamma]].

== ''n''-ary function: function of several variables ==

Functions in applications are often '''functions of several variables''': the values they take depend on a number of different factors. From a mathematical point of view all the variables must be made explicit in order to have a functional relationship - no 'hidden' factors are allowed. Then again, from the mathematical point of view, there is no qualitative difference between functions of one and of several variables. A function of three réal variables is just a function that applies to triples of réal numbers. The following paragraph says this in more formal language.

If the domain of a function is a subset of the [[Cartesian product]] of <var>n</var> sets then the function is called an ''<var>n</var>-ary function''.
For example, the relation ''dist'' has the domain '''R'''&nbsp;×&nbsp;'''R''' and is therefore a [[binary function]].
In that case ''dist''((<var>x</var>,<var>y</var>)) is simply written as ''dist''(<var>x</var>,<var>y</var>).

Another name applied to some types of functions of several variables is [[operation]]. In [[abstract algebra]], operators such as "*" are defined as binary functions; when we write a formula such as ''x''*''y'' in this context, we are implicitly invoking the function *(''x'',''y''), but writing it in a convenient [[infix]] notation.

An important théoretical paradigm, [[functional programming]], takes the function concept as central. In that setting, the handling of '''functions of several variables''' becomes an operational matter, for which the [[lambda calculus]] provides the basic [[syntax]]. The composition of functions (see under '''composing functions''' immediately below) becomes a question of explicit forms of [[substitution]], as used in the [[substitution rule]] of [[calculus]]. In particular, a formalism called [[currying]] can be used to reduce ''n''-ary functions to functions of a single variable.

== Composing functions ==

The functions <var>f</var>:&nbsp;<var>X</var>&nbsp;→&nbsp;<var>Y</var> and <var>g</var>:&nbsp;<var>Y</var>&nbsp;→&nbsp;<var>Z</var> can be ''composed'' by first applying <var>f</var> to an argument <var>x</var> and then applying <var>g</var> to the result.
Thus one obtains a function <var>g</var>&nbsp;<small>o</small>&nbsp;<var>f</var>: <var>X</var>&nbsp;→&nbsp;<var>Z</var> defined by (<var>g</var>&nbsp;<small>o</small>&nbsp;<var>f</var>)(<var>x</var>)&nbsp;:= <var>g</var>(<var>f</var>(<var>x</var>)) for all <var>x</var> in <var>X</var>.
As an example, suppose that an airplane's height at time <var>t</var> is given by the function <var>h</var>(<var>t</var>) and that the oxygen concentration at height <var>x</var> is given by the function <var>c</var>(<var>x</var>).
Then (<var>c</var>&nbsp;<small>o</small>&nbsp;<var>h</var>)(<var>t</var>) describes the oxygen concentration around the plane at time <var>t</var>.

In the mid-[[20th century]], some mathematicians decided that writing "''g''<small>o</small>''f''" to méan "first apply ''f'', then apply ''g''" was too confusing and decided to change notations. They wrote "''xf''" for "''f''(''x'')" and "''xfg''" for "''g''(''f''(''x''))". However, this movement never caught on, and nowadays this notation is found only in old books.

[[Gambar:FNGraph_screenshot.png|ka|jmpl|Various functions graphed using the FNGraph application]]

The functions <var>g</var> and <var>f</var> are [[commutative]] if <var>g</var>&nbsp;<small>o</small>&nbsp;<var>f</var>=<var>f</var>&nbsp;<small>o</small>&nbsp;<var>g</var>.
If
<var>Y</var>⊂<var>X</var>
then 
<var>f</var> may compose with itself; this
is sometimes denoted <var>f</var><sup>&nbsp;2</sup>. (Do not confuse it with the notation
commonly seen in 
[[trigonometric identity|trigonometry]].) 
The '''functional powers'''
<var>f</var> <small>o</small><var>f<sup>&nbsp;n</sup></var> 
=&nbsp;<var>f<sup>&nbsp;n</sup></var> <small>o</small> <var>f</var>
=&nbsp;<var>f</var><sup>&nbsp;<var>n</var>+1</sup>
for
[[natural number|natural]] 
<var>n</var>
follow immediately. This is a generalized version of the common ''f''<sup>−1</sup> for an [[inverse function]]. On their heels comes the idéa of '''functional root''';
given <var>f</var> and <var>n</var>, find a <var>g</var> such that
<var>g<sup>n</sup>=f</var>. 
([[Richard Feynman]] 
illustrated practical use of functional roots in one of his anecdotal books.
&lt;which?&gt;
Tasked with building an 
[[analog computer|analogue]]
[[Trigonometric function|arctan]] 
computer and finding its parts overstressed, he instéad designed a machine for a functional root &lt;fifth?&gt; of arctan and chained enough copies to maké the arctan machine.)

=== Inverse function ===
If a function ''f'':''X''→''Y'' is [[bijective]] then preimages of any element ''y'' in the codomain ''Y'' is a singleton. A function taking ''y''∈''Y'' to its preimage ''f''<sup>−1</sup>(''y'') is a well-defined function called the '''[[Inverse function|inverse]]''' of ''f'' and is denoted by ''f''<sup>−1</sup>.

An example of an inverse function, for ''f''(''x'') = 2''x'', is ''f''(''x'')<sup>−1</sup> = ''x''/2. The inverse function is the function that "undoes" its original. See also [[inverse image]].

Inverses are sometimes difficult or impossible to find. Consider ''f''(''x'') = ''x''<sup>2</sup>. The function ''f''(''x'') = √''x'' is not an inverse when the domain of ''f'' is ''R''. (As -2<sup>2</sup> is 4, but √4 is either 2 or -2).

=== Restrictions and extensions ===
Suppose that <math>X</math> is a [[subset]] of <math>Y</math> and that

:<math>f:Y\rightarrow Z</math>

is a function. Let

:<math>i:X\hookrightarrow Y</math>

be the [[inclusion function]]

:<math>i(x)=x</math>

for ''x ∈ X''.

The ''restriction'' of <math>f</math> to <math>X</math> is then the function <math>f|X = f \circ i</math>. Intuitively, this is the same function as <math>f</math> except that we restrict the domain of <math>f</math> to <math>X</math>.

An ''extension'' of a function <math>g:X\to Z</math> is a function <math>f:Y\to Z</math> defined on a [[superset]] <math>Y</math> of <math>X</math> such that <math>f|X=g</math>. Provided the [[Function domain|domain]] of <math>g</math> is not the [[universal set]], <math>g</math> always has lots of extensions.

== Pointwise operations ==

If <var>f</var>:&nbsp;<var>X</var>&nbsp;→&nbsp;'''R''' and <var>g</var>:&nbsp;<var>X</var>&nbsp;→&nbsp;'''R''' are functions with common domain <var>X</var> and codomain is a [[ring (mathematics)|ring]] '''R''', then one can define the sum function <var>f</var>&nbsp;+&nbsp;<var>g</var>: <var>X</var>&nbsp;→&nbsp;'''R''' and the product function <var>f</var>&nbsp;×&nbsp;<var>g</var>: <var>X</var>&nbsp;→&nbsp;'''R''' as follows:
:(<var>f</var>&nbsp;+&nbsp;<var>g</var>)(<var>x</var>)&nbsp;:= <var>f</var>(<var>x</var>)&nbsp;+&nbsp;<var>g</var>(<var>x</var>);
:(<var>f</var>&nbsp;×&nbsp;<var>g</var>)(<var>x</var>)&nbsp;:= <var>f</var>(<var>x</var>)&nbsp;×&nbsp;<var>g</var>(<var>x</var>);
for all <var>x</var> in <var>X</var>.

This turns the set of all such functions into a ring. The binary operations in that ring have as domain ordered pairs of functions, and as codomain functions. This is an example of climbing up in abstraction, to functions of more complex types.

By taking some other [[abstract algebra|algebraic structure]] <var>A</var> in the place of '''R''', we can turn the set of all functions from <var>X</var> to <var>A</var> into an algebraic structure of the same type in an analogous way.

== Computable and non-computable functions ==

The number of [[computable]] functions from integers to integers is [[countable]], because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the [[cardinality]] of the [[real number]]s. This argument shows that there are functions from integers to integers that are not computable. For examples of noncomputable functions, see the articles on the [[halting problem]] and [[Rice's theorem]].

== Functions from the categorical viewpoint ==

In the context of [[category theory]], a function no longer represents a rule for taking an input to an output, but instéad represents a relationship between its domain and its codomain. Since these functions are no longer functions in the usual sense, they are usually referred to as [[morphism]]s. A morphism is then an ordered triple (''X'', ''Y'', ''f''), where ''f'' is a "function" with domain ''X'' and codomain ''Y''. Since ''X'' and ''Y'' do not necessarily correspond to a set of objects, however, morphisms do not always behave like functions, and, for example, enlarging the codomain (which does nothing to a function) gives a different morphism which you cannot identify with the original one.

Ordinary functions are sometimes referred to as morphisms when they are morphisms in a [[concrete category]].

== References ==
* [http://archives.math.utk.edu/visual.calculus/ Visual Calculus] by [[Lawrence S. Husch]], [[University of Tennessee]] ([[2001]])

== External links ==
* [http://functions.wolfram.com/ http://functions.wolfram.com], a compendium of formulae for and visualizations of mathematical functions
* [http://math.hws.edu/xFunctions/ xFunctions] is a versatile Java applet for exploring functions graphically. It can be used on line or downloaded for use off line.

[[Kategori:Set theory]]