what are functions?
Here is a beginner's guide to functions:
Functions versus Relations
This is not true. The standard definition of function does NOT require "one to one" nor "onto". A function, from set A to set B, is a set of ordered pairs, (x, y), with x from A, y from B, such that we do not have two distinct ordered pairs with the same first element- that is, we cannot have (x, y1), (x, y2) with the same x but different y1 and y2. In more common terms that means that a function assigns a value of y, from set B, to every x in set A such that we do NOT assign different values of y to the same x.
Here's what Wikipedia says: http://en.wikipedia.org/wiki/Function_(mathematics)
Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input.
If not, then it is a multivalued-function. Multivalued function - Wikipedia, the free encyclopedia
My definition corresponds to, as wikipedia calls it, a well-defined function then.
Sorry, but, no, it doesn't. Yours corresponds to "invertible function". The function, for example, $\displaystyle f(x)= x^2$, is a "well defined function", with one and only one output to a particular input but is NOT "one to one and onto" because it does not asociate one and only one input with a particular output.