Functions (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Function_(mathematics)


00:00:20 1 Definition
00:00:41 1.1 Relational approach
00:01:02 1.2 As an element of a Cartesian product over a domain
00:01:33 2 Notation
00:02:04 2.1 Functional notation
00:02:24 2.2 Arrow notation
00:02:55 2.3 Index notation
00:03:37 2.4 Dot notation
00:03:57 2.5 Specialized notations
00:04:18 3 Other terms
00:04:39 3.1 Map
00:04:49 3.2 Morphism
00:05:10 4 Specifying a function
00:05:31 4.1 By listing function values
00:05:51 4.2 By a formula
00:06:12 4.3 Inverse and implicit functions
00:06:53 4.4 Using differential calculus
00:07:24 4.5 By recurrence
00:07:45 5 Representing a function
00:08:06 5.1 Graphs and plots
00:08:26 5.2 Tables
00:08:57 5.3 Bar chart
00:09:18 6 General properties
00:09:39 6.1 Standard functions
00:09:59 6.2 Function composition
00:10:20 6.3 Image and preimage
00:10:41 6.4 Injective, surjective and bijective functions
00:11:12 6.5 Restriction and extension
00:12:14 7 Multivariate function
00:13:06 8 In calculus
00:14:08 8.1 Real function
00:14:49 8.2 Vector-valued function
00:15:10 9 Function space
00:15:51 10 Multi-valued functions
00:16:12 11 In the foundations of mathematics and set theory
00:16:33 12 In computer science
00:17:04 13 See also
00:17:24 13.1 Subpages
00:17:45 13.2 Generalizations
00:17:55 13.3 Related topics
00:18:06 14 Notes
00:18:16 15 References



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SUMMARY
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In mathematics, a function is a relation between sets that associates to every element of a first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X,  the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).
A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means to illustrate the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.