Functions are one of mathematical "work horses". As their name suggests, functions do a lot of work in mathematics. Fundamentally, the concept of mathematical functions is based on the concept of mathematical relationships between quantities.
Ancient civilizations had a fairly good understanding of mathematical relationships. The Babylonian and Egyptian civilizations provided the earliest records of mathematical relationships between quantities. For example, the Babylonians devised tables of compound interest, squares, cubes, square roots and cube roots. They were also able to solve quadratic equations and problems involving ten unknowns in ten equations.
The Egyptians had formulas they used to determine wages, taxes, areas of fields and volumes of granaries. They were also able to calculate the number of bricks they needed for specific structures and the amount of grain required to produce a specific quantity of beer.
In ancient Greece, Greek mathematicians such as Eudoxus, the Pythagoreans, Euclid and the group of Greek mathematicians known as the great Alexandrian mathematicians (Eratosthenes, Archimedes, Appollonius, Ptolemy, Diophantus and Hipparchus) used mathematical relationships to produce great works. For example, Archimedes stated and proved many theorems on complex areas and volumes among which was the proof that the volume of a sphere is two-thirds the volume of the cylinder that circumscribe the sphere. Archimedes was also a great physicist and in this capacity invented the compound pulley and the basic law of hydrostatics known as Archimedes Principle, which states that the weight lost by a body immersed in a fluid is equal to the weight of the fluid it displaces.
Ancient Indian mathematicians such as Aryabata and Brahmagupta (to name a few) also showed a strong knowledge of mathematical relationships. For example, Aryhabata studied indeterminate equations and found solutions to many indeterminate equations. Brahmagupta who was the first mathematician to use negative numbers, formulated many mathematical relationships with positive and negative numbers which were used in commerce.
Arab mathematician al Khwarizmi, undoubtedly one of the fathers of algebra, wrote one of the first systematic explanations of algebra. In his treatise, al jabr (a term from which algebra was derived), al-Khwarizmi presented many examples and proofs of algebraic relationships. Other Arab mathematicians such as Al-karaji and Alhazen also contributed to the development of mathematical relationships. The point is that mathematical relationships upon which the concept of functions is based have their origins in antiquity, and ancient mathematicians from diverse cultures helped developed this concept.
The great Italian physicist and astronomer Galileo Galilei (A.D 1564 - 1642) is generally credited with pioneering the transition of the general concept of mathematical relationships into the specialized concept of functions. In his pioneering works on scientific methods, Galileo was very explicit in his expressions of the dependency of one quantity on another quantity in the problems he formulated and in the statements of the results of its experiments. For example, in stating one of the results of its experiment on falling bodies, Galileo wrote: "the times of descent along inclined planes of the same height but of differnt slopes, are to each other as the lenghts of these slopes." However, Galileo did not use the term function in his works.
In 1637, French philosopher and mathematician Rene Descartes (A.D. 1596-1650) became the first person to mathematically use the term function to describe a variable x raised to power n (xn). Then in 1694, German mathematician Gottfried Leibniz (A.D. 1646- 1716) used the term to describe certain characteristics of curves, such as the slope of a curve at a specific point. About the middle of the 18th century, Swiss mathematician Leonhard Euler (A.D.1707-1783) described a function as an expression or formula involving various arguments. Euler also introduced the notation f(x) in 1734.
The definition of function closest to the modern definition of function was independently presented by German mathematicians Peter Dirichlet, Karl Weiestrass (A.D.1815-1897) and Russian mathematician Nicholay Lobachevski (A.D. 1798-1856). Dirichlet described a function as a variable y, called a dependent variable whose value is determined according to its specific relationship to a variable x, called the independent variable.
What is the nature of the special mathematical relationships known as functions? One of the key characteristics of functions is expressed in Dirichlet definition and it is this: the mathematical relationships that are functions are between dependent variables and independent variables. Another key characteristic is the one-one mapping between the dependent variable and the independent variables. This is the single most important determinant of whether or not a mathematical relationship is a function.