Function (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-01-11

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Function_(mathematics)


00:02:18 1 Definition
00:05:59 1.1 Relational approach
00:09:17 2 Notation
00:11:26 2.1 Functional notation
00:15:25 2.2 Arrow notation
00:19:42 2.3 Index notation
00:21:59 2.4 Dot notation
00:23:53 2.5 Specialized notations
00:24:56 3 Specifying a function
00:31:25 4 Representing a function
00:31:32 4.1 Graphs and plots
00:31:43 4.2 Tables
00:32:06 4.3 Bar chart
00:35:07 5 General properties
00:36:48 5.1 Canonical functions
00:37:29 5.2 Equality of functions
00:37:49 5.3 Function composition
00:39:52 5.4 Image and preimage
00:42:08 5.5 Injective, surjective and bijective functions
00:42:42 5.6 Restriction and extension
00:45:27 6 Multivariate function
00:48:39 7 In calculus
00:56:05 7.1 Real function
01:04:13 7.2 Complex function
01:10:22 7.3 Function of several real or complex variables
01:15:50 7.4 Vector-valued function
01:17:12 8 Function space
01:23:29 9 Generalizations
01:25:00 9.1 Natural extension
01:28:28 9.2 Multi-valued functions
01:29:34 10 In foundations of mathematics and set theory
01:30:38 11 In computer science
01:30:48 11.1 Lambda calculus
01:33:17 12 See also
01:37:52 12.1 Subpages
01:39:47 12.2 Generalizations
01:42:05 12.3 Related topics
01:43:54 13 Notes
01:44:03 14 References
01:44:12 15 Further reading
01:44:21 16 External links



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.7742796116178983
Voice name: en-GB-Wavenet-B


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics, a function was 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 its graph which is the set of all pairs (x, f (x)). When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere. For example, graphs of functions are commonly used in newspapers for representing the evolution of price indexes and stock market indexes
Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are "the central objects of investigation" in most fields of mathematics.