Editing Function (mathematics) (section)

== In calculus ==
{{further|History of the function concept}}

The idea of function, starting in the 17th century, was fundamental to the new [[infinitesimal calculus]]. At that time, only [[real-valued function|real-valued]] functions of a [[function of a real variable|real variable]] were considered, and all functions were assumed to be [[smooth function|smooth]]. But the definition was soon extended to [[#Multivariate function|functions of several variables]] and to [[functions of a complex variable]]. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.

Functions are now used throughout all areas of mathematics. In introductory [[calculus]], when the word ''function'' is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with [[STEM]] majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as [[real analysis]] and [[complex analysis]].

=== Real function ===
{{see also|Real analysis}}
[[File:Gerade.svg|thumb|right|Graph of a linear function]]
[[File:Polynomialdeg2.svg|thumb|right|Graph of a polynomial function, here a quadratic function.]]
[[File:Sine cosine one period.svg|thumb|right|Graph of two trigonometric functions: [[sine]] and [[cosine]].]]
A ''real function'' is a [[real-valued function|real-valued]] [[function of a real variable]], that is, a function whose codomain is the [[real number|field of real numbers]] and whose domain is a set of [[real number]]s that contains an [[interval (mathematics)|interval]]. In this section, these functions are simply called ''functions''.

The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are [[continuous function|continuous]], [[differentiable function|differentiable]], and even [[analytic function|analytic]]. This regularity insures that these functions can be visualized by their [[#Graph and plots|graphs]]. In this section, all functions are differentiable in some interval.

Functions enjoy [[pointwise operation]]s, that is, if {{mvar|f}} and {{mvar|g}} are functions, their sum, difference and product are functions defined by

<math display="block">\begin{align}
(f+g)(x)&=f(x)+g(x)\\
(f-g)(x)&=f(x)-g(x)\\
(f\cdot g)(x)&=f(x)\cdot g(x)\\
\end{align}.</math>

The domains of the resulting functions are the [[set intersection|intersection]] of the domains of {{mvar|f}} and {{mvar|g}}. The quotient of two functions is defined similarly by

<math display="block">\frac fg(x)=\frac{f(x)}{g(x)},</math>

but the domain of the resulting function is obtained by removing the [[zero of a function|zeros]] of {{mvar|g}} from the intersection of the domains of {{mvar|f}} and {{mvar|g}}.

The [[polynomial function]]s are defined by [[polynomial]]s, and their domain is the whole set of real numbers. They include [[constant function]]s, [[linear function]]s and [[quadratic function]]s. [[Rational function]]s are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid [[division by zero]]. The simplest rational function is the function <math>x\mapsto \frac 1x,</math> whose graph is a [[hyperbola]], and whose domain is the whole [[real line]] except for 0.

The [[derivative]] of a real differentiable function is a real function. An [[antiderivative]] of a continuous real function is a real function that has the original function as a derivative. For example, the function <math display="inline">x\mapsto\frac 1x</math> is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for {{math|1=''x'' = 1}}, is a differentiable function called the [[natural logarithm]].

A real function {{mvar|f}} is [[monotonic function|monotonic]] in an interval if the sign of <math>\frac{f(x)-f(y)}{x-y}</math> does not depend of the choice of {{mvar|x}} and {{mvar|y}} in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function {{mvar|f}} is monotonic in an interval {{mvar|I}}, it has an [[inverse function]], which is a real function with domain {{math|''f''(''I'')}} and image {{mvar|I}}. This is how [[inverse trigonometric functions]] are defined in terms of [[trigonometric functions]], where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a [[bijection]] between the real numbers and the positive real numbers. This inverse is the [[exponential function]].

Many other real functions are defined either by the [[implicit function theorem]] (the inverse function is a particular instance) or as solutions of [[differential equation]]s. For example, the [[sine]] and the [[cosine]] functions are the solutions of the [[linear differential equation]]

<math display="block">y''+y=0</math>

such that

<math display="block">\sin 0=0, \quad \cos 0=1, \quad\frac{\partial \sin x}{\partial x}(0)=1, \quad\frac{\partial \cos x}{\partial x}(0)=0.</math>

=== Vector-valued function ===
{{main|Vector-valued function|Vector field}}

When the elements of the codomain of a function are [[vector (mathematics and physics)|vectors]], the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its [[velocity vector]] is a vector-valued function.

Some vector-valued functions are defined on a subset of <math>\mathbb{R}^n</math> or other spaces that share geometric or [[topological]] properties of <math>\mathbb{R}^n</math>, such as [[manifolds]]. These vector-valued functions are given the name ''vector fields''.