Function (mathematics)

Any subset of the Cartesian product of two sets ${\displaystyle X}$ and ${\displaystyle Y}$ defines a binary relation ${\displaystyle R\subseteq X\times Y}$ between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function, given above.

A univalent or functional relation is a relation such that

${\displaystyle (x,y)\in R\;\land \;(x,z)\in R\implies y=z}$

for all ${\displaystyle x}$ in ${\displaystyle X}$ and ${\displaystyle y,z}$ in ${\displaystyle Y}$. Univalent relations may be identified with functions with codomain ${\displaystyle Y,}$ whose domain is a subset of X.

A total relation is a relation such that

${\displaystyle \forall x\in X\;\exists y\in Y:(x,y)\in R.}$

Formally, functions are identified as relations that are both univalent and total. Univalent relations that are not total are called partial functions.

Various properties of functions and function composition may be reformulated in the language of relations. For example, a function is injective if the converse relation ${\displaystyle R^{\text{T}}\subseteq Y\times X}$ is univalent, where the converse relation is defined as ${\displaystyle R^{\text{T}}=\{(y,x)\mid (x,y)\in R\}.}$[6]

The set of all functions from some given domain to a codomain is sometimes identified with the Cartesian product of copies of the codomain, indexed by the domain. Namely, given sets ${\displaystyle X}$ and ${\displaystyle Y,}$ any function ${\displaystyle f\colon X\to Y}$ is an element of the Cartesian product of copies of ${\displaystyle Y}$s over the index set ${\displaystyle X}$

${\displaystyle f\in \prod _{X}Y=Y^{X}.}$

Viewing ${\displaystyle f}$ as tuple with coordinates, then for each ${\displaystyle x\in X}$, the ${\displaystyle x}$th coordinate of this tuple is the value ${\displaystyle f(x)\in Y.}$ This reflects the intuition that for each ${\displaystyle x\in X,}$ the function picks some element ${\displaystyle y\in Y,}$ namely, ${\displaystyle f(x).}$ (This point of view is used for example in the discussion of a choice function.)

Infinite Cartesian products are often simply "defined" as sets of functions.[7]

As first used by Leonhard Euler in 1734,[8] functions are denoted by a symbol consisting generally of a single letter in italic font, most often the lower-case letters f, g, h. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). By convention, in this case, a roman type is used, such as "sin" for the sine function, in contrast to italic font for single-letter symbols.

The notation (read: "y equals f of x")

${\displaystyle y=f(x)}$

means that the pair (x, y) belongs to the set of pairs defining the function f. If X is the domain of f, the set of pairs defining the function is thus, using set-builder notation,

${\displaystyle \{(x,f(x)):x\in X\}.}$

Often, a definition of the function is given by what f does to the explicit argument x. For example, a function f can be defined by the equation

${\displaystyle f(x)=\sin(x^{2}+1)}$

for all real numbers x. In this example, f can be thought of as the composite of several simpler functions: squaring, adding 1, and taking the sine. However, only the sine function has a common explicit symbol (sin), while the combination of squaring and then adding 1 is described by the polynomial expression ${\displaystyle x^{2}+1}$. In order to explicitly reference functions such as squaring or adding 1 without introducing new function names (e.g., by defining function g and h by ${\displaystyle g(x)=x^{2}}$ and ${\displaystyle h(x)=x+1}$), one of the methods below (arrow notation or dot notation) could be used.

Sometimes the parentheses of functional notation are omitted when the symbol denoting the function consists of several characters and no ambiguity may arise. For example, ${\displaystyle \sin x}$ can be written instead of ${\displaystyle \sin(x).}$

For explicitly expressing domain X and the codomain Y of a function f, the arrow notation is often used (read: "the function f from X to Y" or "the function f mapping elements of X to elements of Y"):

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

or

${\displaystyle X~{\stackrel {f}{\to }}~Y.}$

This is often used in relation with the arrow notation for elements (read: "f maps x to f (x)"), often stacked immediately below the arrow notation giving the function symbol, domain, and codomain:

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

For example, if a multiplication is defined on a set X, then the square function ${\displaystyle \operatorname {sqr} }$ on X is unambiguously defined by (read: "the function ${\displaystyle \operatorname {sqr} }$ from X to X that maps x to x ⋅ x")

${\displaystyle {\begin{aligned}\operatorname {sqr} \colon X&\to X\\x&\mapsto x\cdot x,\end{aligned}}}$

the latter line being more commonly written

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

Often, the expression giving the function symbol, domain and codomain is omitted. Thus, the arrow notation is useful for avoiding introducing a symbol for a function that is defined, as it is often the case, by a formula expressing the value of the function in terms of its argument. As a common application of the arrow notation, suppose ${\displaystyle f\colon X\times X\to Y;\;(x,t)\mapsto f(x,t)}$ is a two-argument function, and we want to refer to a partially applied function ${\displaystyle X\to Y}$ produced by fixing the second argument to the value t₀ without introducing a new function name. The map in question could be denoted ${\displaystyle x\mapsto f(x,t_{0})}$ using the arrow notation for elements. The expression ${\displaystyle x\mapsto f(x,t_{0})}$ (read: "the map taking x to ${\displaystyle f(x,t_{0})}$") represents this new function with just one argument, whereas the expression ${\displaystyle f(x_{0},t_{0})}$ refers to the value of the function f at the point ${\displaystyle (x_{0},t_{0})}$.

Index notation is often used instead of functional notation. That is, instead of writing f (x), one writes ${\displaystyle f_{x}.}$

This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element ${\displaystyle f_{n}}$ is called the nth element of sequence.

The index notation is also often used for distinguishing some variables called parameters 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 ${\displaystyle x\mapsto f(x,t)}$ (see above) would be denoted ${\displaystyle f_{t}}$ using index notation, if we define the collection of maps ${\displaystyle f_{t}}$ by the formula ${\displaystyle f_{t}(x)=f(x,t)}$ for all ${\displaystyle x,t\in X}$.

In the notation ${\displaystyle x\mapsto f(x),}$ the symbol x does not represent any value, it is simply a placeholder meaning that, if 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, x may be replaced by any symbol, often an interpunct " ⋅ ". This may be useful for distinguishing the function f (⋅) from its value f (x) at x.

For example, ${\displaystyle a(\cdot )^{2}}$ may stand for the function ${\displaystyle x\mapsto ax^{2}}$, and ${\displaystyle \textstyle \int _{a}^{\,(\cdot )}f(u)\,du}$ may stand for a function defined by an integral with variable upper bound: ${\displaystyle \textstyle x\mapsto \int _{a}^{x}f(u)\,du}$.

There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis, linear forms and the vectors they act upon are denoted using a dual pair to show the underlying duality. This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.

A function is often also called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular map is often used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to H instead of group homomorphism from G to H). Some authors[16] reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function.

Some authors, such as Serge Lang,[17] use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions.

In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.

Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.

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. E.g., if ${\displaystyle A=\{1,2,3\}}$, then one can define a function ${\displaystyle f\colon A\to \mathbb {R} }$ by ${\displaystyle f(1)=2,f(2)=3,f(3)=4.}$

Functions are often defined by a formula 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, ${\displaystyle f}$ can be defined by the formula ${\displaystyle f(n)=n+1}$, for ${\displaystyle n\in \{1,2,3\}}$.

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 zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} ,}$ 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, ${\displaystyle f(x)={\sqrt {1+x^{2}}}}$ defines a function ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ whose domain is ${\displaystyle \mathbb {R} ,}$ because ${\displaystyle 1+x^{2}}$ is always positive if x is a real number. On the other hand, ${\displaystyle f(x)={\sqrt {1-x^{2}}}}$ defines a function from the reals to the reals whose domain is reduced to the interval [–1, 1]. (In old texts, such a domain was called the domain of definition of the function.)

Functions are often classified by the nature of formulas that can that define them:

• A quadratic function is a function that may be written ${\displaystyle f(x)=ax^{2}+bx+c,}$ where a, b, c are 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 integers. For example, ${\displaystyle f(x)=x^{3}-3x-1,}$ and ${\displaystyle f(x)=(x-1)(x^{3}+1)+2x^{2}-1.}$
• A rational function is the same, with divisions also allowed, such as ${\displaystyle f(x)={\frac {x-1}{x+1}},}$ and ${\displaystyle f(x)={\frac {1}{x+1}}+{\frac {3}{x}}-{\frac {2}{x-1}}.}$
• An algebraic function is the same, with nth roots and roots of polynomials also allowed.
• An elementary function[note 5] is the same, with logarithms and exponential functions allowed.

A function ${\displaystyle f\colon X\to Y,}$ with domain X and codomain Y, is bijective, if for every y in Y, there is one and only one element x in X such that y = f(x). In this case, the inverse function of f is the function ${\displaystyle f^{-1}\colon Y\to X}$ that maps ${\displaystyle y\in Y}$ to the element ${\displaystyle x\in X}$ such that y = f(x). For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It has these an inverse, called the exponential function that maps the real numbers onto the positive numbers.

If a function ${\displaystyle f\colon X\to Y}$ is not bijective, it may occur that one can select subsets ${\displaystyle E\subseteq X}$ and ${\displaystyle F\subseteq Y}$ such that the restriction of f to E is a bijection from E to 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 [0, π] onto the interval [–1, 1], and its inverse function, called arccosine, maps [–1, 1] onto [0, π]. The other inverse trigonometric functions are defined similarly.

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

For example, the equation of the unit circle ${\displaystyle x^{2}+y^{2}=1}$ defines a relation on real numbers. If –1 < x < 1 there are two possible values of y, one positive and one negative. For x = ± 1, these two values become both equal to 0. Otherwise, there is no possible value of y. This means that the equation defines two implicit functions with domain [–1, 1] and respective codomains [0, +∞) and (–∞, 0].

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

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

Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of 1/x that is 0 for x = 1. Another common example is the error function.

More generally, many functions, including most special functions, can be defined as solutions of differential equations. 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 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 ${\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}}$. 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 numbers, 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 and the trigonometric functions of a complex number.

Functions whose domain are the nonnegative integers, known as sequences, are often defined by recurrence relations.

The factorial function on the nonnegative integers (${\displaystyle n\mapsto n!}$) is a basic example, as it can be defined by the recurrence relation

${\displaystyle n!=n(n-1)!\quad {\text{for}}\quad n>0,}$

and the initial condition

${\displaystyle 0!=1.}$

The function mapping each year to its US motor vehicle death count, shown as a line chart

The same function, shown as a bar chart

Given a function ${\displaystyle f\colon X\to Y,}$ its graph is, formally, the set

${\displaystyle G=\{(x,f(x)):x\in X\}.}$

In the frequent case where X and Y are subsets of the real numbers (or may be identified with such subsets, e.g. intervals), an element ${\displaystyle (x,y)\in G}$ may be identified with a point having coordinates x, y in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a plot that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the graph of the function. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function

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

consisting of all points with coordinates ${\displaystyle (x,x^{2})}$ for ${\displaystyle x\in \mathbb {R} ,}$ yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function ${\displaystyle x\mapsto x^{2},}$ with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates ${\displaystyle (r,\theta )=(x,x^{2}),}$ the plot obtained is Fermat's spiral.

A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function ${\displaystyle f\colon \{1,\ldots ,5\}^{2}\to \mathbb {R} }$ defined as ${\displaystyle f(x,y)=xy}$ can be represented by the familiar multiplication table

On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:

Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.

Bar charts are often used for representing functions whose domain is a finite set, the natural numbers, or the integers. In this case, an element x of the domain is represented by an interval of the x-axis, and the corresponding value of the function, f(x), is represented by a rectangle whose base is the interval corresponding to x and whose height is f(x) (possibly negative, in which case the bar extends below the x-axis).

There are a number of standard functions that occur frequently:

• For every set X, there is a unique function, called the empty function from the empty set to X. The existence of the empty function is a convention that is needed for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements.
• For every set X and every singleton set {s}, there is a unique function from X to {s}, which maps every element of X to s. This is a surjection (see below) unless X is the empty set.
• Given a function ${\displaystyle f\colon X\to Y,}$ the canonical surjection of f onto its image ${\displaystyle f(X)=\{f(x)\mid x\in X\}}$ is the function from X to f(X) that maps x to f(x).
• For every subset A of a set X, the inclusion map of A into X is the injective (see below) function that maps every element of A to itself.
• The identity function on a set X, often denoted by id_X, is the inclusion of X into itself.

Given two functions ${\displaystyle f\colon X\to Y}$ and ${\displaystyle g\colon Y\to Z}$ such that the domain of g is the codomain of f, their composition is the function ${\displaystyle g\circ f\colon X\rightarrow Z}$ defined by

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

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

The composition ${\displaystyle g\circ f}$ is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both ${\displaystyle g\circ f}$ and ${\displaystyle f\circ g}$ satisfy these conditions, the composition is not necessarily commutative, that is, the functions ${\displaystyle g\circ f}$ and ${\displaystyle f\circ g}$ need not be equal, but may deliver different values for the same argument. For example, let f(x) = x² and g(x) = x + 1, then ${\displaystyle g(f(x))=x^{2}+1}$ and ${\displaystyle f(g(x))=(x+1)^{2}}$ agree just for ${\displaystyle x=0.}$

The function composition is associative in the sense that, if one of ${\displaystyle (h\circ g)\circ f}$ and ${\displaystyle h\circ (g\circ f)}$ is defined, then the other is also defined, and they are equal. Thus, one writes

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

The identity functions ${\displaystyle \operatorname {id} _{X}}$ and ${\displaystyle \operatorname {id} _{Y}}$ are respectively a right identity and a left identity for functions from X to Y. That is, if f is a function with domain X, and codomain Y, one has ${\displaystyle f\circ \operatorname {id} _{X}=\operatorname {id} _{Y}\circ f=f.}$

• 
  A composite function g(f(x)) can be visualized as the combination of two "machines".
• 
  A simple example of a function composition
• 
  Another composition. In this example, (g ∘ f )(c) = #.

Let ${\displaystyle f\colon X\to Y.}$ The image by f of an element x of the domain X is f(x). If A is any subset of X, then the image of A by f, denoted f(A) is the subset of the codomain Y consisting of all images of elements of A, that is,

${\displaystyle f(A)=\{f(x)\mid x\in A\}.}$

The image of f is the image of the whole domain, that is f(X). It is also called the range of f, although the term may also refer to the codomain.[18]

On the other hand, the inverse image, or preimage by f of a subset B of the codomain Y is the subset of the domain X consisting of all elements of X whose images belong to B. It is denoted by ${\displaystyle f^{-1}(B).}$ That is

${\displaystyle f^{-1}(B)=\{x\in X\mid f(x)\in B\}.}$

For example, the preimage of {4, 9} under the square function is the set {−3,−2,2,3}.

By definition of a function, the image of an element x of the domain is always a single element of the codomain. However, the preimage of a single element y, denoted ${\displaystyle f^{-1}(y),}$ may be empty or contain any number of elements. For example, if f is the function from the integers to themselves that maps every integer to 0, then ${\displaystyle f^{-1}(0)=\mathbb {Z} }$.

If ${\displaystyle f\colon X\to Y}$ is a function, A and B are subsets of X, and C and D are subsets of Y, then one has the following properties:

• ${\displaystyle A\subseteq B\Longrightarrow f(A)\subseteq f(B)}$
• ${\displaystyle C\subseteq D\Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)}$
• ${\displaystyle A\subseteq f^{-1}(f(A))}$
• ${\displaystyle C\supseteq f(f^{-1}(C))}$
• ${\displaystyle f(f^{-1}(f(A)))=f(A)}$
• ${\displaystyle f^{-1}(f(f^{-1}(C)))=f^{-1}(C)}$

The preimage by f of an element y of the codomain is sometimes called, in some contexts, the fiber of y under f.

If a function f has an inverse (see below), this inverse is denoted ${\displaystyle f^{-1}.}$ In this case ${\displaystyle f^{-1}(C)}$ may denote either the image by ${\displaystyle f^{-1}}$ or the preimage by f of C. This is not a problem, as these sets are equal. The notation ${\displaystyle f(A)}$ and ${\displaystyle f^{-1}(C)}$ may be ambiguous in the case of sets that contain some subsets as elements, such as ${\displaystyle \{x,\{x\}\}.}$ In this case, some care may be needed, for example, by using square brackets ${\displaystyle f[A],f^{-1}[C]}$ for images and preimages of subsets, and ordinary parentheses for images and preimages of elements.

Let ${\displaystyle f\colon X\to Y}$ be a function.

The function f is injective (or one-to-one, or is an injection) if f(a) ≠ f(b) for any two different elements a and b of X. Equivalently, f is injective if, for any ${\displaystyle y\in Y,}$ the preimage ${\displaystyle f^{-1}(y)}$ contains at most one element. An empty function is always injective. If X is not the empty set, and if, as usual, Zermelo–Fraenkel set theory is assumed, then f is injective if and only if there exists a function ${\displaystyle g\colon Y\to X}$ such that ${\displaystyle g\circ f=\operatorname {id} _{X},}$ that is, if f has a left inverse. If f is injective, for defining g, one chooses an element ${\displaystyle x_{0}}$ in X (which exists as X is supposed to be nonempty),[note 6] and one defines g by ${\displaystyle g(y)=x}$ if ${\displaystyle y=f(x),}$ and ${\displaystyle g(y)=x_{0}}$, if ${\displaystyle y\not \in f(X).}$

The function f is surjective (or onto, or is a surjection) if the range equals the codomain, that is, if f(X) = Y. In other words, the preimage ${\displaystyle f^{-1}(y)}$ of every ${\displaystyle y\in Y}$ is nonempty. If, as usual, the axiom of choice is assumed, then f is surjective if and only if there exists a function ${\displaystyle g\colon Y\to X}$ such that ${\displaystyle f\circ g=\operatorname {id} _{Y},}$ that is, if f has a right inverse. The axiom of choice is needed, because, if f is surjective, one defines g by ${\displaystyle g(y)=x,}$ where ${\displaystyle x}$ is an arbitrarily chosen element of ${\displaystyle f^{-1}(y).}$

The function f is bijective (or is bijection or a one-to-one correspondence) if it is both injective and surjective. That is f is bijective if, for any ${\displaystyle y\in Y,}$ the preimage ${\displaystyle f^{-1}(y)}$ contains exactly one element. The function f is bijective if and only if it admits an inverse function, that is a function ${\displaystyle g\colon Y\to X}$ such that ${\displaystyle g\circ f=\operatorname {id} _{X},}$ and ${\displaystyle f\circ g=\operatorname {id} _{Y}.}$ (Contrarily to the case of surjections, this does not require the axiom of choice.)

Every function ${\displaystyle f\colon X\to Y}$ may be factorized as the composition i ∘ s of a surjection followed by an injection, where s is the canonical surjection of X onto f(X), and i is the canonical injection of f(X) into Y. This is the canonical factorization of f.

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "f maps X onto Y" differs from "f maps X into B" in that the former implies that f is surjective, while the latter makes no assertion about the nature of f the mapping. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage to be more symmetrical.

If ${\displaystyle f\colon X\to Y}$ is a function and S is a subset of X, then the restriction of ${\displaystyle f}$ to S, denoted ${\displaystyle f|_{S}}$, is the function from S to Y defined by

${\displaystyle f|_{S}(x)=f(x)}$

for all x in S. Restrictions can be used to define partial inverse functions: if there is a subset S of the domain of a function ${\displaystyle f}$ such that ${\displaystyle f|_{S}}$ is injective, then the canonical surjection of ${\displaystyle f|_{S}}$ onto its image ${\displaystyle f|_{S}(S)=f(S)}$ is a bijection, and thus has an inverse function from ${\displaystyle f(S)}$ to S. One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval [0, π]. The image of this restriction is the interval [–1, 1], and thus the restriction has an inverse function from [–1, 1] to [0, π], which is called arccosine and is denoted arccos.

Function restriction may also be used for "gluing" functions together. Let ${\displaystyle \textstyle X=\bigcup _{i\in I}U_{i}}$ be the decomposition of X as a union of subsets, and suppose that a function ${\displaystyle f_{i}\colon U_{i}\to Y}$ is defined on each ${\displaystyle U_{i}}$ such that for each pair ${\displaystyle i,j}$ of indices, the restrictions of ${\displaystyle f_{i}}$ and ${\displaystyle f_{j}}$ to ${\displaystyle U_{i}\cap U_{j}}$ are equal. Then this defines a unique function ${\displaystyle f\colon X\to Y}$ such that ${\displaystyle f|_{U_{i}}=f_{i}}$ for all i. This is the way that functions on manifolds are defined.

An extension of a function f is a function g such that f is a restriction of g. A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.

Here is another classical example of a function extension that is encountered when studying homographies of the real line. A homography is a function ${\displaystyle h(x)={\frac {ax+b}{cx+d}}}$ such that ad – bc ≠ 0. Its domain is the set of all real numbers different from ${\displaystyle -d/c,}$ and its image is the set of all real numbers different from ${\displaystyle a/c.}$ If one extends the real line to the projectively extended real line by including ∞, one may extend h to a bijection from the extended real line to itself by setting ${\displaystyle h(\infty )=a/c}$ and ${\displaystyle h(-d/c)=\infty }$.

Graph of a linear function

Graph of a polynomial function, here a quadratic function.

Graph of two trigonometric functions: sine and cosine.

A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an 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, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.

Functions enjoy pointwise operations, that is, if f and g are functions, their sum, difference and product are functions defined by

${\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(f-g)(x)&=f(x)-g(x)\\(f\cdot g)(x)&=f(x)\cdot g(x)\\\end{aligned}}.}$

The domains of the resulting functions are the intersection of the domains of f and g. The quotient of two functions is defined similarly by

${\displaystyle {\frac {f}{g}}(x)={\frac {f(x)}{g(x)}},}$

but the domain of the resulting function is obtained by removing the zeros of g from the intersection of the domains of f and g.

The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions 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 ${\displaystyle x\mapsto {\frac {1}{x}},}$ 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 is differentiable in any open interval in which the original function is continuous. For example, the function ${\displaystyle x\mapsto {\frac {1}{x}}}$ is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for x = 1, is a differentiable function called the natural logarithm.

A real function f is monotonic in an interval if the sign of ${\displaystyle {\frac {f(x)-f(y)}{x-y}}}$ does not depend of the choice of x and 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 f is monotonic in an interval I, it has an inverse function, which is a real function with domain f(I) and image 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 equations. For example, the sine and the cosine functions are the solutions of the linear differential equation

${\displaystyle y''+y=0}$

such that

${\displaystyle \sin 0=0,\quad \cos 0=1,\quad {\frac {\partial \sin x}{\partial x}}(0)=1,\quad {\frac {\partial \cos x}{\partial x}}(0)=0.}$

When the elements of the codomain of a function are 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 ${\displaystyle \mathbb {R} ^{n}}$ or other spaces that share geometric or topological properties of ${\displaystyle \mathbb {R} ^{n}}$, such as manifolds. These vector-valued functions are given the name vector fields.