Function (mathematics)

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In mathematics, the function of is initially an idealization of how quantity varies depending on other quantities. For example, the position of the planet is a function of time . Historically, this concept was expounded by a very small calculus at the end of the seventeenth century, and, until the 19th century, functions were deemed differentiated (ie they had a high degree of order). The concept of function has been formalized in the late 19th century in terms of set theory, and this has broadened the domain of applying the concept.

Function is a process or relation that links every element of x from a set of X , domain of a function, to a single element y from another set Y (maybe the same set), codomain of the function. If the function is called f , it is not < >) (read f from x ), element x is argument or input of the function, and y is the value of the function, output , or the image of x by f . The symbol used to represent the input is a function variable (it is often said that f is a function of the x variable).

A function is represented by the graph which is the set of all pairs ( x , f ( x )) . When domains and codomains are collections of numbers, each pair can be considered as Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called a function graph. This is a useful functional representation, commonly used everywhere, for example in newspapers.

Functions are widely used in science, and in most areas of mathematics. Their very important role has been said that they are "the main object of inquiry" in most areas of mathematics.

Video Function (mathematics)

Definisi

When the first student starts learning the function, intuitive ideas such as "rules" and "peers" are used. In more advanced mathematics, a more precise definition is needed. The exact mathematical definition of a function depends on the idea of ​​the ordered pair.

A function f from one set X to a set Y is determined by a set of G of the ordered pair ( x , y ), like that x ? X , y ? Y , and each element X is the first component of exactly one ordered pair in G . In other words, for every x in X there is exactly one element y , such that the spelled ( x , y ) is included in the set of pairs that define the function < i> f .

In this definition, X and Y are each called domains and > codomain of the f function. If ( x , y ) is included in the set that defines f , then y is image x below f , or the value of f applied to the argument x . Especially in the context of numbers, someone also says that y is the value of f for value x of the variable , or, still shorter, y is the value f of x , denoted as y = f ( x ) .

Formally, the function definition includes fixing its domain and its codomain. However, this is not always explicitly given, and, without some (possibly difficult) calculations, people only know that the domains are in a larger set. Typically, this occurs in a mathematical analysis, where "the function of X to Y " refers frequently to a function that may have a proper subset of X as domain. For example, the function of from real to real can refer to the real-valued function of the real variable, and this sentence does not mean that the domain of the function is the whole set of real numbers, but only that domain is a set of real numbers contains open intervals that are not empty. For example, if f is a function that has a real number as a domain and a codomain, then the function, maps the value of x to ${\ displaystyle g (x) = {\ tfrac {1} {f (x)}}}$ is a function of g from real to real, whose domain is a collection of real x , such f ( x )? 0 . In many cases, the exact domain is difficult to determine, but it is rarely a problem to work with such functions.

The range of functions is the set of images of all elements in a domain. However, the range is sometimes used as a synonym of codomain, generally in old textbooks.

Maps Function (mathematics)

Notation

There are various standard ways to show the function. Particularly they, involving the explicit denotation of their variables, give rise to a fine point, often polished in the care of basic functions: functions are different from their values. Thus, the function f must be distinguished from its value f ( x ₀ ) on the value x ₀ on its domain. To some extent, even a working mathematician will link the two in an informal setting for convenience, and to avoid using harsh language. However, strictly speaking, it is not sufficient to say "let ${\ displaystyle f: \ mathbb {R} \ to \ mathbb {R}}$ becomes the function f ( x ) = x ² ", because f ( x ) and x ² both can be understood as value .Instead, it's true, though long-winded, to say "leave ${\ displaystyle f: \ mathbb {R} \ to \ mathbb {R}}$ becomes a function defined by the equation f ( x ) = x < ² , is valid for all real values ​​ x ".

Functional notation

To work with certain functions, it is useful to use symbols to symbolize them. This symbol generally consists of one letter in italics, most commonly lowercase f , g , h . Some of the widely used functions are represented by symbols consisting of several letters (generally the abbreviation of their name). In this case, the name is set in a romance type, such as " sin " for the sine function.

Notasi (baca: and same dengan f dari x )

${\ displaystyle y = f (x)} Â Â$

berarti bahwa pasangan ( x , y ) termasuk dalam kumpulan pasangan yang mendefinisikan fungsi f . Jika X adalah domain f , kumpulan pasangan yang mendefinisikan fungsi demikian, menggunakan notasi set-builder,

${\ displaystyle \ {(x, f (x)) \ titik dua x \ dalam X \}.}  ÂÂ$

Kadang-kadang tanda kurung dari notasi fungsional dihilangkan, ketika simbol yang menunjukkan fungsi terdiri dari beberapa karakter dan tidak ada ambiguitas yang mungkin timbul, adapun

${\ displaystyle \ without x \ quad {\ text {bukannya}} \ quad \ sin (x).} Â Â$

Notasi panah

Untuk membuat eksplisit domain X dan codomain Y dari fungsi f notasi panah sering digunakan (baca: f memetakan X ke Y ):

${\ displaystyle f \ colon X \ to Y}  ÂÂ$

atau

${\ displaystyle X ~ {\ stackrel {f} {\ ke}} ~ Y.} Â Â$

Ini sering digunakan sehubungan dengan notasi panah untuk elemen (baca: f peta x ke f ( x ) ):

${\ displaystyle x \ mapsto f (x).}  ÂÂ$

This is useful for avoiding symbol recognition for defined functions, as is often the case, with a formula that states the value of a function in terms of its argument. For example, if multiplication is defined on a set of X , then the square function in X is clearly defined by (read: function of X to X that maps x to x ? x )

${\ displaystyle {\ begin {aligned} & amp; X \ to X \\ & amp; x \ mapsto x \ cdot x, \ end {aligned}}}$

bar terakhir lebih sering ditulis

${\ displaystyle x \ mapsto x ^ 2.} Â Â$

Notasi indeks

Notasi indeks sering digunakan sebagai pengganti notasi fungsional. Yaitu, f ( x ) , seseorang menulis ${\ displaystyle f_ {x}.} Â Â$

Ini biasanya berlaku untuk fungsi yang domainnya adalah himpunan bilangan asli. Fungi seperti ini disebut urutan, dan, dalam hal ini elemen ${\ displaystyle f_ {n}} Â$ disebut sebagai elemen urutan n .

Index notation is also often used to distinguish some variables called parameters from "true variables". In fact, the parameter is a special variable that is considered fixed during the study of the problem.

Point notation

Dalam notasi ${\ displaystyle x \ mapsto f (x),} Â$ symbol x tidak mewakili nilai apa pun, itu hanyalah sebuah placeholder yang berarti bahwa, jika x Adalah diganti dengan nilai apa pun di sebelah kiri panah, itu harus diganti dengan nilai yang sama di sebelah kanan panah. Oleh karena itu, x dapat digantikan oleh simbol apa pun, sering kali berupa interpunct "? ". Ini mungkin berguna untuk membedakan fungsi f (?) dari nilainya f ( x ) to x .

Misalnya, ${\ displaystyle a (\ cdot) ^ 2}} Â$ dapat berdiri untuk ${\ displaystyle x \ mapsto ax ^ {2}} Â$ , give $Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âa\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}^{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}}$              (               ?               )                               f           (           u           )          d           u                     {\ displaystyle \ textstyle \ int _ {a} ^ {\, (\ cdot)} f (u) du}  dapat berdiri untuk fungsi yang didefinisikan oleh integral dengan variabel batas atas: ${\ displaystyle \ textstyle x \ mapsto \ int _ {a} ^ {x} f (u) du}  Â$ .

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Menentukan fungsi

By definition of a function, a particular function, in general, is defined by connecting to each domain element at least one element of its codomain. When domain and codomain are a collection of numbers, this association can take the form of computation taken as input of any element of the domain and generate as output in the codomain. This computation can be explained by the formula. (This is the starting point of algebra, which replaces many numerical calculations similar to a formula that describes this calculation by a formula involving variables, representing the calculated input as an unspecified number). The type of definition of this function often uses the auxiliary function that is defined previously.

Misalnya, fungsi f dari real that real, ditentukan oleh rumus ${\ displaystyle f (x) = {\ sqrt {1 x2}},} Â$ mempekerjakan, sebagai fungsi tambahan, fungsi kuadrat (memetakan semua real que real non-negatif), fungsi akar kuadrat (memetakan sumber daya non-negatif that real non -negatif), under royal bilangan penambahan. Seluruh himpunan bilangan real dapat diambil sebagai domain f , bahkan jika domain dari fungsi akar kuadrat dibatasi ke bilangan real yang tidak negatif; gambar f terdiri dari real yang tidak kurang dari satu.

Komputasi yang memungkinkan mendefinisikan fungsi sering dapat dijelaskan oleh suatu algoritma, dan segala jenis algoritma dapat digunakan. Terkadang, definisi fungsi dapat melibatkan elemen atau properti yang dapat didefinisikan, tetapi tidak dihitung. Sebagai contoh, jika seseorang menganggap set ${\ displaystyle {\ mathcal {P}}}  ÂÂ$ dari program dalam bahasa pemrograman tertentu, yang mengambil integer sebagai input. Fungsi terminating adalah fungsi yang mengembalikan 1 jika program ${\ displaystyle {\ mathcal {P}}}  ÂÂ$ berjalan selamanya ketika dieksekusi pada input bilangan bulat yang diberikan, dan mengembalikan 0 sebaliknya. Ini adalah teorema dasar teori computability yang tidak bisa ada algoritma untuk menghitung fungsi ini. Lebih umum, teori computability adalah studi tentang fungsi komputasi, yaitu fungsi yang dapat dihitung oleh suatu algoritma.

The above ways define the function of defining "in sequence", that is, each value is defined independently of other values. This is not necessarily the case.

When the domain of a function is a set of non-negative integers or, more commonly, when the domain is ordered properly, the function can be defined by induction or recursion, which means (approx) that the calculation of the value of the function for some given input requires the value of function for lower input. For example, the Fibonacci sequence is a function of natural numbers into themselves defined by two initial values ​​and a formula, which repeats the two preceding arguments (see above for the use of indexes for function arguments):

_{         F                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                          =         0         ,                         F              Â 1                          =         1         ,                            and                                 F                 Â ·                           =                 F                 Â ·      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ, -   Â 1                                          F                 Â ·      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ, -    Â 2                                         Â   for                  n         & gt;         1.               {\ displaystyle F_ {0} = 0, \ quad F_ {1} = 1, \ quad {\ text {and}} \ quad F_ {n} = F_ {n -1} F_ {n-2} \ quad {\ text {for}} n & gt; 1.}  Â}

In calculus, the function under consideration has a wide order. That is, the value of a function at a point corresponds to the function values ​​at adjacent points. This allows defining it with functional equations (for example, the gamma function is a unique merpaticfiction such as ${\ displaystyle \ Gamma (1) = 1}$ , and ${\ displaystyle \ Gamma (z 1) = z \ Gamma (z)}$ for any z complex that is not a non-positive integer), with differential equations (for example, natural logarithm is the solution of equation differential ${\ displaystyle {\ frac {d {\ xn}} {dx}} = {\ frac {1} {x}}}$ ln (1) = 0 ), integral equations or analytic continuation.

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Represents the function

Because the function may be a complex object, it is often useful to draw a function graph to gain a global view of its properties. Some functions may also represent a histogram

Graph

Diberikan fungsi ${\ displaystyle f: X \ to Y,} Â Â$ grafik nya adalah, secara resmi, himpunan

${\ displaystyle G = \ {(x, y): x \ dalam X {\ text {and}} y = f (x) \}.} Â Â$

Dalam kasus yang sering terjadi di mana X dan Y adalah himpunan bagian dari bilangan real (atau dapat diidentifikasi ke subset tersebut), sebuah elemen ${\ displaystyle (x, y) \ di G} Â Â$ dapat diidentifikasi dengan titik koordinat x , and di Pesawat Cartesian. Menandai titik-titik ini menyediakan gambar, umumnya kurva, yang juga disebut grafik dari fungsi . Misalnya grafik fungsi persegi

${\ displaystyle x \ mapsto x ^ 2}} Â Â$

adalah parabola yang terdiri dari semi titik koordinat ${\ displaystyle (x, x ^ 2)} Â$ untuk ${\ displaystyle x \ in \ mathbb {R}.} Â Â$

It is possible to effectively draw a graph of a function only if its function is sufficiently organized, that is, whether the function can be differentiated (or divisible) or if its domain can be identified by an integer or a subset of integers.

Jika salah satu domain atau codomain dari fungsi adalah subkumpulan ${\ displaystyle \ mathbb {R} ^ {n},} Â$ graphik adalah bagian dari ruang Cartesian dimensi yang lebih tinggi, dan berbagai teknik telah dikembangkan untuk menggambarnya, termasuk penggunaan warna untuk mewakili salah satu dimensi.

Histogram

Histograms are often used to represent functions whose domain is limited, or are real numbers or integers. In this case, the x element of the domain is represented by the interval of x -axis, and a point ( x , y ) of the graph is represented by a rectangle with an interval basis corresponding to x and height y .

In statistics, histograms are often used to represent very irregular functions. For example, to represent a function that associates its weight with each member of some population, one draws a histogram of a function corresponding to each weight interval of the number of persons, whose weight is included in this interval.

There are many variants of this method, see Histogram for details.

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General properties

In this section, we describe the general nature of the function, which does not depend on certain properties of the domain and codomain.

Canonical Functions

Some functions are uniquely determined by their domain and codomain, and are sometimes called canonical :

• For each set of X , there is a unique function, called empty function from the empty set to X . This function is not interesting by itself, but it is useful for simplifying statements, equal to empty numbers (equal to 0) and empty products equal to 1.
• For each set of X and every single set { s } , there is a unique function, called canonical surjection from X to { s } , which maps to s each element X . This is a surjection (see below), unless X is an empty set.
• Given the function ${\ displaystyle f \ colon X \ to Y,}$ canonical surjection of f on image ${\ displaystyle f (X) = \ {f (x) \ mid x \ in X \}}$ is a function of X to f ( X ) that maps x to f ( x )
• For each X section of a set of Y , canonical injection from X to Y is a syringe function (see below) that maps each element X to itself.
• The identity function X , often denoted by $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          id                 Â                                {\ displaystyle \ operatorname {id} _ {X}}} is a canonical X injector into itself.

Composition function

Diberikan dua fungsi ${\ displaystyle f: X \ to Y} Â$ dan ${\ displaystyle g: Y \ to Z} Â$ sehingga domain g adalah codomain f , komposisi mereka adalah fungsi ${\ displaystyle g \ circ f \ colon X \ rightarrow Z} Â$ ditentukan oleh

${\ displaystyle (g \ circ f) (x) = g (f (x)).} Â Â$

That is, the value of ${\ displaystyle g \ circ f}$ is obtained by first applying f to x to get y = f ( x ) and then apply g on the y result to get g ( y ) = g ( f ( x )) . In the first applied function notation is always written on the right.

Komposisi ${\ displaystyle g \ circ f} Â$ adalah operasi pada fungsi yang ditentukan hanya jika codomain dari fungsi pertama adalah domain dari yang kedua. Bahkan ketika ${\ displaystyle g \ circ f} Â$ dan ${\ displaystyle f \ circ g} Â Â$ keduanya didefinisikan, komposisi tidak komutatif. Misalnya, jika f ( x ) = x ² dan < i> g ( x ) = x 1 , seseorang memilize ${\ displaystyle g (f (x)) = x ^ 2 1 \ neq f (g (x)) = (x 1) 2 }. } Â Â$

Komposisi fungsi adalah asosiatif dalam arti bahwa, jika salah satu dari ${\ displaystyle (h \ circ g) \ circ f} Â$ dan ${\ displaystyle h \ circ (g \ circ f)} Â$ didefinisikan, maka yang lain juga didefinisikan, dan mereka sama. Jadi, seseorang menulis