The concept of function is one of the most important in mathematics. However, its history is relatively short. M. Kline credits [Kline, p. 338] Galileo (1564-1642) with the first statements of dependency of one quantity on another, e.g., "The times of descent along inclined planes of the same height, but of different slopes, are to each other as the lengths of these slopes." In a 1673 manuscript Leibniz used the word "function" to mean any quantity varying from point to point of a curve, like the length of the tangent or the normal. The curve itself was said to be given by an equation. But in 1714, he already used the word "function" to mean quantities that depend on a variable. The notation f(x) was introduced by Euler in 1734. Still, in the 1930s, a well known Russian mathematician N. Luzin wrote:

Functions, especially of the numeric variety, are often confused with formulas by means of which they are defined. In one of the discrete mathematics textbooks, the authors fling a particularly inept remark to the effect that "Whereas classical mathematics is about formulas, discrete mathematics is as much about algorithms as about formulas." Charitably, I interpret the maxim as the authors' attempt to emphasize the importance of functions in mathematics in general and discrete mathematics in particular. In their view, I believe, the efficiency of function computations gains prominence when it comes to practical matters. In mathematics, the function of two variables f(x, y) = x² - y² can be equally well defined as f(x, y) = (x - y)(x + y). In algorithmic mathematics there is an important difference between the two definitions: one requires two multiplications and one addition (with the sign minus), the other needs one multiplication and two additions. The latter is faster! But the authors, of course, might have had their own reasons.

• Functional Notations and Terminology
• Examples of Functions
• Historical Introduction
• Addition of Functions
  • Addition and Subtraction of Functions
• Multiplication of Functions

References

1. M. Kline, Mathematical Thought From Ancient to Modern Times I, Oxford University Press, 1972
2. N. Luzin, Function, Mathematical Evolutions, A. Shenitzer and J. Stillwell (eds.), MAA, 2002