Editing Function (mathematics) (section)

== Notation ==

There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below.

=== Functional notation ===
The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letter {{mvar|f}}. Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in 

<math display="block">f(x), \quad \sin(3),\quad \text{or}\quad f(x^2+1).</math>

The argument between the parentheses may be a [[variable (mathematics)|variable]], often {{mvar|x}}, that represents an arbitrary element of the domain of the function, a specific element of the domain ({{math|3}} in the above example), or an [[expression (mathematics)|expression]] that can be evaluated to an element of the domain (<math>x^2+1</math> in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let <math>f(x)=\sin(x^2+1)</math>".

When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write {{math|sin ''x''}} instead of {{math|sin(''x'')}}.

Functional notation was first used by [[Leonhard Euler]] in 1734.<ref>{{cite book|first1=Ron|last1=Larson|first2=Bruce H.|last2=Edwards|title=Calculus of a Single Variable|page=19|year=2010|publisher=Cengage Learning|isbn=978-0-538-73552-0}}</ref> Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a [[roman type]] is customarily used instead, such as "{{math|sin}}" for the [[sine function]], in contrast to italic font for single-letter symbols.

The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let <math>f(x)</math> be a function". This is an [[abuse of notation]] that is useful for a simpler formulation.

=== Arrow notation ===
Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced "[[maps to]]". For example, <math>x\mapsto x+1</math> is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of <math>\R</math> is implied.

The domain and codomain can also be explicitly stated, for example:

<math display="block">\begin{align}
\operatorname{sqr}\colon \Z &\to \Z\\
x &\mapsto x^2.\end{align}</math>

This defines a function {{math|sqr}} from the integers to the integers that returns the square of its input.

As a common application of the arrow notation, suppose <math>f: X\times X\to Y;\;(x,t) \mapsto f(x,t)</math> is a function in two variables, and we want to refer to a [[Partial application|partially applied function]] <math>X\to Y</math> produced by fixing the second argument to the value {{math|''t''<sub>0</sub>}} without introducing a new function name.  The map in question could be denoted <math>x\mapsto f(x,t_0)</math> using the arrow notation. The expression <math>x\mapsto f(x,t_0)</math> (read: "the map taking {{mvar|x}} to {{mvar|f}} of {{mvar|x}} comma {{mvar|t}} nought") represents this new function with just one argument, whereas the expression {{math|''f''(''x''<sub>0</sub>, ''t''<sub>0</sub>)}} refers to the value of the function {{mvar|f}} at the {{nowrap|point {{math|(''x''<sub>0</sub>, ''t''<sub>0</sub>)}}.}}

=== Index notation ===

Index notation may be used instead of functional notation. That is, instead of writing {{math|''f''{{hair space}}(''x'')}}, one writes <math>f_x.</math>

This is typically the case for functions whose domain is the set of the [[natural number]]s. Such a function is called a [[sequence (mathematics)|sequence]], and, in this case the element <math>f_n</math> is called the {{mvar|n}}th element of the sequence.

The index notation can also be used for distinguishing some variables called ''[[Parameter (mathematics)|parameter]]s'' from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem.  For example, the map <math>x\mapsto f(x,t)</math> (see above) would be denoted <math>f_t</math> using index notation, if we define the collection of maps <math>f_t</math> by the formula <math>f_t(x)=f(x,t)</math> for all <math>x,t\in X</math>.

=== Dot notation ===

In the notation
<math>x\mapsto f(x),</math>
the symbol {{mvar|x}} does not represent any value; it is simply a [[placeholder name|placeholder]], meaning that, if {{mvar|x}} is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, {{mvar|x}} may be replaced by any symbol, often an [[interpunct]] "{{math| ⋅ }}". This may be useful for distinguishing the function {{math|''f''{{hair space}}(⋅)}} from its value {{math|''f''{{hair space}}(''x'')}} at {{mvar|x}}.

For example, <math> a(\cdot)^2</math> may stand for the function <math> x\mapsto ax^2</math>, and <math display="inline"> \int_a^{\, (\cdot)} f(u)\,du</math> may stand for a function defined by an [[integral]] with variable upper bound: <math display="inline"> x\mapsto \int_a^x f(u)\,du</math>.

=== Specialized notations ===
There are other, specialized notations for functions in sub-disciplines of mathematics.  For example, in [[linear algebra]] and [[functional analysis]], [[linear form]]s and the [[Vector (mathematics and physics)|vectors]] they act upon are denoted using a [[dual pair]] to show the underlying [[Duality (mathematics)|duality]].  This is similar to the use of [[bra–ket notation]] in quantum mechanics.  In [[Mathematical logic|logic]] and the [[theory of computation]], the function notation of [[lambda calculus]] is used to explicitly express the basic notions of function [[Abstraction (computer science)|abstraction]] and [[Function application|application]].  In [[category theory]] and [[homological algebra]], networks of functions are described in terms of how they and their compositions [[Commutative property|commute]] with each other using [[commutative diagram]]s that extend and generalize the arrow notation for functions described above.

===Functions of more than one variable===
In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function {{mvar|f}} can be defined as mapping any pair of real numbers <math>(x, y)</math> to the sum of their squares, <math>x^2 + y^2</math>. Such a function is commonly written as <math>f(x, y)=x^2 + y^2</math> and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as <math>f(w,x, y)</math>, <math>f(w,x, y, z)</math>.