The modern period began at the end of the sixteenth century. During this time functions became equivalent to analytical expressions. This revolutionized mathematics. Eventually by the eighteenth century the idea of a function as an analytical expression was insufficient, and a more generalized definition was developed. Essentially "a function y of the variable x, y = f(x), is a relation between pairs of elements of two number sets, X and Y, such that to each element x from the first set X one and only one element y from the second set Y is assigned according to some definite rule," as stated by 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.). This rule can be shown using words, an x, y table, an analytic expression, a graph, or other means.
During the late sixteenth century and early seventeenth century more computations were being done, the use of symbols was growing, and the inclusion of imaginary and complex numbers into the number system further developed the function concept. Function concepts now took place between sets of numbers and not just quantities. Logarithms were discovered and trigonometry was advancing. Using signs for math operations and relations (+, -, powers, =, etc.) and signs for unknowns and knowns let them write down algebraic equations and expressions in a symbolic form.
Science, especially mechanics, was developing. New concepts of the laws of nature and their relationship to numerous quantities was growing. The chief problem of science at the time was the relationship between curvilinear motion and forces affecting motion.
The use of formulas to introduce functions was starting. Descartes developed the idea of introducing a function analytically in his Geometry (1637). He wanted to reduce the solution of all algebraic problems and equations to some standard procedure for constructing their real roots. Descartes was the first to make it clear that an equation in x and y is a way to show a dependence between variable quantities so that the values of one of them can be calculated from the corresponding values of the other one. Descartes was also the first to classify various algebraic curves.
In the middle of the seventeenth century a discovery was made independently by Pietro Mengoli, James Gregory, and Issac Newton. They made it possible to write any functional relation analytically. They figured out how to develop functions into infinite power series. Other infinite expressions of functions were added later.