When many people think of functions, they think of something like
but functions have a broader definition than expressions in x. A function is actually a one to one mapping between sets. The sets can be almost anything, but most frequently we are concerned with sets of number, and functions which numbers from one set onto numbers from another set.
A function maps elements of one set, called the domain, onto elements of another set, called the codomain.
The function
below operates on the set
with
and
While more than one element of the domain must map to a single element of the codomain (think of:
andis a function) if one element of a set maps onto the same element of another set, the mapping cannot be a function. The mapping below does not define a function, since the element
is mapped to the elements1 and 4 by the mapping.
does not define a function, since
or