Editing Function (mathematics) (section)

== Specifying a function ==
Given a function <math>f</math>, by definition, to each element <math>x</math> of the domain of the function <math>f</math>, there is a unique element associated to it, the value <math>f(x)</math> of <math>f</math> at <math>x</math>. There are several ways to specify or describe how <math>x</math> is related to <math>f(x)</math>, both explicitly and implicitly. Sometimes, a theorem or an [[axiom]] asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function <math>f</math>.

=== By listing function values ===
On a finite set a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. For example, if <math>A = \{ 1, 2, 3 \}</math>, then one can define a function <math>f: A \to \mathbb{R}</math> by <math>f(1) = 2, f(2) = 3, f(3) = 4.</math>

=== By a formula ===
Functions are often defined by an [[expression (mathematics)|expression]] that describes a combination of [[arithmetic operations]] and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain.
For example, in the above example, <math>f</math> can be defined by the formula <math>f(n) = n+1</math>, for <math>n\in\{1,2,3\}</math>.

When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the [[zero of a function|zeros]] of auxiliary functions. Similarly, if [[square root]]s occur in the definition of a function from <math>\mathbb{R}</math> to <math>\mathbb{R},</math> the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.

For example, <math>f(x)=\sqrt{1+x^2}</math> defines a function <math>f: \mathbb{R} \to \mathbb{R}</math> whose domain is <math>\mathbb{R},</math> because <math>1+x^2</math> is always positive if {{mvar|x}} is a real number. On the other hand, <math>f(x)=\sqrt{1-x^2}</math> defines a function from the reals to the reals whose domain is reduced to the interval {{closed-closed|−1, 1}}. (In old texts, such a domain was called the ''domain of definition'' of the function.)

Functions can be classified by the nature of formulas that define them:
* A [[quadratic function]] is a function that may be written <math>f(x) = ax^2+bx+c,</math> where {{math|''a'', ''b'', ''c''}} are [[constant (mathematics)|constants]].
* More generally, a [[polynomial function]] is a function that can be defined by a formula involving only additions, subtractions, multiplications, and [[exponentiation]] to nonnegative integer powers. For example, <math>f(x) = x^3-3x-1</math> and <math>f(x) = (x-1)(x^3+1) +2x^2 -1</math> are polynomial functions of <math>x</math>.
* A [[rational function]] is the same, with divisions also allowed, such as <math>f(x) = \frac{x-1}{x+1},</math> and <math>f(x) = \frac 1{x+1}+\frac 3x-\frac 2{x-1}.</math>
* An [[algebraic function]] is the same, with [[nth root|{{mvar|n}}th roots]] and [[zero of a function|roots of polynomials]] also allowed.
* An [[elementary function]]<ref group=note>Here "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree.</ref> is the same, with [[logarithm]]s and [[exponential functions]] allowed.

=== Inverse and implicit functions ===
A function <math>f : X\to Y,</math> with domain {{mvar|X}} and codomain {{mvar|Y}}, is [[bijective]], if for every {{mvar|y}} in {{mvar|Y}}, there is one and only one element {{mvar|x}} in {{mvar|X}} such that {{math|1=''y'' = ''f''(''x'')}}. In this case, the [[inverse function]] of {{mvar|f}} is the function <math>f^{-1} : Y \to X</math> that maps <math>y\in Y</math> to the element <math>x\in X</math> such that {{math|1=''y'' = ''f''(''x'')}}. For example, the [[natural logarithm]] is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the [[exponential function]], that maps the real numbers onto the positive numbers.

If a function <math>f: X\to Y</math> is not bijective, it may occur that one can select subsets <math>E\subseteq X</math> and <math>F\subseteq Y</math> such that the [[restriction of a function|restriction]] of {{mvar|f}} to {{mvar|E}} is a bijection from {{mvar|E}} to {{mvar|F}}, and has thus an inverse. The [[inverse trigonometric functions]] are defined this way. For example, the [[cosine function]] induces, by restriction, a bijection from the [[interval (mathematics)|interval]] {{closed-closed|0, ''π''}} onto the interval {{closed-closed|−1, 1}}, and its inverse function, called [[arccosine]], maps {{closed-closed|−1, 1}} onto {{closed-closed|0, ''π''}}. The other inverse trigonometric functions are defined similarly.

More generally, given a [[binary relation]] {{mvar|R}} between two sets {{mvar|X}} and {{mvar|Y}}, let {{mvar|E}} be a subset of {{mvar|X}} such that, for every <math>x\in E,</math> there is some <math>y\in Y</math> such that {{math|''x R y''}}. If one has a criterion allowing selecting such a {{mvar|y}} for every <math>x\in E,</math> this defines a function <math>f: E\to Y,</math> called an [[implicit function]], because it is implicitly defined by the relation {{mvar|R}}.

For example, the equation of the [[unit circle]] <math>x^2+y^2=1</math> defines a relation on real numbers. If {{math|−1 < ''x'' < 1}} there are two possible values of {{mvar|y}}, one positive and one negative. For {{math|1=''x'' = ± 1}}, these two values become both equal to 0. Otherwise, there is no possible value of {{mvar|y}}. This means that the equation defines two implicit functions with domain {{closed-closed|−1, 1}} and respective codomains {{closed-open|0, +∞}} and {{open-closed|−∞, 0}}.

In this example, the equation can be solved in {{mvar|y}}, giving <math>y=\pm \sqrt{1-x^2},</math> but, in more complicated examples, this is impossible. For example, the relation <math>y^5+y+x=0</math> defines {{mvar|y}} as an implicit function of {{mvar|x}}, called the [[Bring radical]], which has <math>\mathbb R</math> as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and [[nth root|{{mvar|n}}th roots]].

The [[implicit function theorem]] provides mild [[differentiability]] conditions for existence and uniqueness of an implicit function in the neighborhood of a point.

=== Using differential calculus ===
Many functions can be defined as the [[antiderivative]] of another function. This is the case of the [[natural logarithm]], which is the antiderivative of {{math|1/''x''}} that is 0 for {{math|1=''x'' = 1}}. Another common example is the [[error function]].

More generally, many functions, including most [[special function]]s, can be defined as solutions of [[differential equation]]s. The simplest example is probably the [[exponential function]], which can be defined as the unique function that is equal to its derivative and takes the value 1 for {{math|1=''x'' = 0}}.

[[Power series]] can be used to define functions on the domain in which they converge. For example, the [[exponential function]] is given by <math display="inline">e^x = \sum_{n=0}^{\infty} {x^n \over n!}</math>. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its [[Taylor series]] in some interval, this power series allows immediately enlarging the domain to a subset of the [[complex number]]s, the [[disc of convergence]] of the series. Then [[analytic continuation]] allows enlarging further the domain for including almost the whole [[complex plane]]. This process is the method that is generally used for defining the [[logarithm]], the [[exponential function|exponential]] and the [[trigonometric functions]] of a complex number.

=== By recurrence ===
{{main|Recurrence relation}}
Functions whose domain are the nonnegative integers, known as [[sequence]]s, are sometimes defined by [[recurrence relation]]s.

The [[factorial]] function on the nonnegative integers (<math>n\mapsto n!</math>) is a basic example, as it can be defined by the recurrence relation

<math display="block">n!=n(n-1)!\quad\text{for}\quad n>0,</math>

and the initial condition

<math display="block">0!=1.</math>