Function (mathematics)

[note 1][4] When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.

(read 'maps to') is used to specify where a particular element x in the domain is mapped to by f. This allows the definition of a function without naming.

In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set.

It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory.

A function with domain X and codomain Y is a binary relation R between X and Y that satisfies the two following conditions:[10] This definition may be rewritten more formally, without referring explicitly to the concept of a relation, but using more notation (including set-builder notation): A function is formed by three sets, the domain

This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain.

is more or less equivalent to the proof or disproof of one of the major open problems in mathematics, the Riemann hypothesis.

A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem).

For example, the position of a car on a road is a function of the time travelled and its average speed.

When using functional notation, one usually omits the parentheses surrounding tuples, writing

[11] 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 "sin" for the sine function, in contrast to italic font for single-letter symbols.

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.

Some authors[14] reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function.

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

the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.

This process is the method that is generally used for defining the logarithm, the exponential and the trigonometric functions of a complex number.

) is a basic example, as it can be defined by the recurrence relation and the initial condition A graph is commonly used to give an intuitive picture of a function.

A bar chart can represent a function 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).

[17][20] If, as usual in modern mathematics, the axiom of choice is assumed, then f is surjective if and only if there exists a function

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.

Here is another classical example of a function extension that is encountered when studying homographies of the real line.

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.

The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic.

Thus one antiderivative, which takes the value zero for x = 1, is a differentiable function called the natural logarithm.

For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.

For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets i for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets −i.

In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions.

In its original form, lambda calculus does not include the concepts of domain and codomain of a function.

Schematic depiction of a function described metaphorically as a "machine" or " black box " that for each input yields a corresponding output
The red curve is the graph of a function , because any vertical line has exactly one crossing point with the curve.
Diagram of a function
Diagram of a relation that is not a function. One reason is that 2 is the first element in more than one ordered pair. Another reason is that neither 3 nor 4 are the first element (input) of any ordered pair therein
A binary operation is a typical example of a bivariate function which assigns to each pair the result .
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
Graph of a linear function
Graph of a polynomial function, here a quadratic function.
Graph of two trigonometric functions: sine and cosine .
Together, the two square roots of all nonnegative real numbers form a single smooth curve.