Johann Bernoulli and Euler:
Function = Arbitrary Analytic Expression

Johann Bernoulli used the word function in a article in 1698 on the solution to a problem involving curves. He was the first to define a function as an analytic expression. He proposed the Greek letter phi be used as notation for a function or phix. Euler introduced f for function and included brackets for f(x).

Even though symbolism and analytic ideas of functions were growing, geometric notions of a function still persisted. Finally in the middle of the nineteenth century things like area of a surface and length of a curve needed to be defined analytically.

Euler's (1748) definition of a function was:

A function of a variable quantity is an analytic expression composed in any way from this variable quantity and numbers or constant quantities.
Like Johann Bernoulli, he sees functions as analytic expressions. He also had definitions for a constant and a variable.

In chapter 4 of his Introductio Euler reduces all analytic functions to one type of function, the power series of the type A + Bz + Cz2 + Dz3 + ···. He challenged anyone to try to disprove this statement. He added that any power of z, not just positive ones, could be used. A function as any analytic expression was accepted and studied by others in the years following this.

After this time there was debate over "mixed" or "discontinuous" functions, now called piecewise functions. There was also controversy about the vibrating string. More information on these can be found in A.P. Youschkevitch's 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.) and in N. Luzin's article "The Evolution of...Function: Part I" (The American Mathematical Monthly, vol. 105 (1998), no. 1, pp. 59-67.).

Eventually Euler modified his definition to: "the general notion of correspondence between pairs of elements each belonging to its own set of values of variable quantities."

His idea of relationship was illustrated in his new abstract definition in 1755 in his Institutiones calculi differentialis:

If some quantities so depend on other quantities that if the latter are changed the former undergo change, then the former quantities are called functions of the latter. This denomination is of broadest nature and compromises every method by means of which one quantity could be determined by others. If, therefore, x denotes a variable quantity, then all quantities which depend on x in any way or are determined by it are called functions of it.
This definition gained wider recognition and use. Other mathematicians like, Condorcet (1778), Lacroix (1797), Fourier (1821), Lobachevsky (1834), and Dirichlet (1837) used and developed further Euler's definition.