The modern concept of the function began to materialize around the seventeenth century when various mathematicians were trying to solve practical problems of motion, such as tracking the moon as a method of sea navigation or keeping precise time. The problems all involved trying to find a certain relation between variables.
Descartes made the distinction between "geometric and mechanical curves" which gave rise to the distinction between "algebraic" and "transcendental" functions that we know today. Leibniz, in a manuscript of 1673, was the first to formally use the term "function." John Bernoulli defined it as a quantity consisting of variables and constants in "any manner," meaning to include both algebraic and transcendental functions. Euler in 1734 was the first to formalize the notation y = f(x).
Source: Kline, M. (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press (New York).
No. There are cases where the inverse might be a function, but this is not guaranteed.
To recall, to get the inverse of a function, the domain and the codomain have to be switched. And a function is any relation in which every element in the domain x corresponds only one element in the codomain y; however, not every element in y has to correspond only to one element in x for the relation to be a function (for example, quadratic equations which appear in graphs as parabolas). When the domain and codomain are switched in such a case, the result will no longer be a function, as then the elements of the new domain will correspond to too many elements in the codomain.
This is also visually demonstrable through graphs via the vertical line test. This test works because the image of non-elliptical functions can never double back upon itself (because each element of x corresponds to only one element of y). Therefore, vertical lines can only pass through the image at one point at most.
Functions are inverted visually by reflecting the function along the line y=x. In the case of parabolas, if reflected along this line, they would immediately fail the vertical line test.