Short answers, pulled from the story.
A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain and the set Y is called the codomain. Functions are used throughout science, engineering, and most fields of mathematics.
The domain is the set X of inputs to which the function assigns values, while the codomain is the set Y that holds the possible outputs. A function pairs each element of the domain with exactly one element of the codomain.
Leonhard Euler first used functional notation in 1734, writing a function's name followed by its argument in parentheses. Functions are often denoted by a letter such as f, g, or h.
A partial function from X to Y assigns at most one output to each input, so some inputs may be undefined. It is an ordinary function whose domain is a subset of X, called the domain of definition. When that domain of definition equals X, the partial function is called a total function.
A function is injective when different inputs always give different outputs, surjective when every codomain element is the output of some input, and bijective when it is both. A bijective function is a one-to-one correspondence and admits an inverse function. These terms were coined as French words in the second quarter of the 20th century by the Bourbaki group.
The first formal definition of a function appeared at the end of the 19th century, stated in terms of set theory. It defines a function as a binary relation, a subset of the Cartesian product of the domain and codomain, where every domain element appears in exactly one pair. This set-theoretic formalization greatly increased the possible applications of the concept.