The concept of a mathematical function

The concept of a mathematical function did not form immediately, but a long and thorny path of development had passed. In the mathematical works of antiquity, the functional dependence is already contained in the first, mathematically expressed relations between the quantities, in the first formulas for finding the area and volume of certain figures. For example, the area formula of the circle S = 3*r2, deduced by Babylonian scholars 4,5 thousand years ago, unconsciously indicates that the area of the circle is a function of its radius. As a way of specifying a function, one can also consider tables of squares and cubes of numbers or trigonometric tables, the compilation of which began long before the beginning of our era. But, only since the 17th century the concept of function becomes one of the most important in the knowledge of the real world.

The concept of a mathematical function did not form immediately, but a long and thorny path of development had passed

In the 17th century, thanks to the penetration of the idea of variables into mathematics, the concept of a mathematical function is fully consciously used, but still intuitive. For example, in the works of Descartes, Fermat, Newton, and Leibniz, the concept of a mathematical function is associated either with geometric or mechanical representations: the ordinates of the points of the curves as a function of the abscissas (x); path and speed as a function of time (t), etc. True, Descartes in his "Geometry" considered only those curves that can be accurately represented using equations, mostly algebraic ones. Thus, the concept of a mathematical function became identified with the concept of an analytic expression, i.e, a formula.

For the first time the word "function" (from the Latin functio - execution, commission) was used by Leibniz in 1673, and the expression "function of X" was used by Leibniz and I. Bernoulli since 1698.

The definition of a mathematical function was first given in 1718 by one of the students and employees of Leibniz, the outstanding Swiss mathematician Johann Bernoulli: "A variable quantity function is a quantity formed by any method from this variable and constant".

New steps in the development of natural science and mathematics in the 19th century. and further generalization of the concept of a mathematical function. A great contribution to the solution of the question about what should be understood as a function was made by Euler, D'Alembert, D. Bernoulli, J. Fourier and other scientists of the 18th century.

Today the concept of a mathematical function can be expressed as follows: a function is a mathematical concept that reflects the relationship between the elements of sets. In other words, a function is a rule that assigns to each element of one set (called the domain of definition) some element of another set (called the range of values).

The concept of a mathematical function expresses an intuitive idea of how one quantity completely determines the value of another quantity. So the value of the variable uniquely determines the value of the expression, and the value of the month uniquely determines the value of the next month. Likewise, some pre-conceived algorithm with varying input data produces certain output data.

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