The historical development of the concept of function has often been avoided, not only by students, but by many scholars of the history of mathematics. The concept is explained in today's math classes and math textbooks, but most don't tell how the concept developed.
There is disagreement over when the concept of function started. Many think it began with Descartes and his use of coordinates and graphing. Others believe it started c. 2000 B.C. with the Babylonians and their tables of various values. Some think it began with the astronomical calculations of ancient scholars. The Greeks used proportions, a type of functional dependence. In the fourteenth century functions were calculated and expressed geometrically. By the middle of the eighteenth century, the general concept of function was an arbitrary relation between pairs of elements, each from its own set. Keep in mind, using symbols to represent variables and operations was not used much until the sixteenth century. There was no abstract, generalized way to describe a function or relation between quantities.
A.P. Youschkevitch, in his article "The Concept of a Function up to the Middle of the 19th Century" (Archives for History of Exact Sciences, vol. 16 (1976/77), pp. 37-85.), classifies the evolution of the function concept into three stages: antiquity, the middle ages, and the modern period. An overview of each is described in the sections listed below along with a description of the function concept today.