TITLE: How do you define functions for non-mathematicians?
QUESTION [36 upvotes]: I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to start with the definition of one, yet many "official" definitions I have found too convoluted (or poorly written) for general use.
Here's one of the better "light" definitions I've found:

A function is a relationship which
  assigns to each input (or domain)
  value, a unique output (or range)
  value."

This sounds simple enough on the surface, but putting myself "in the head" of a student makes me pause. It's almost too compact with potentially ambiguous words for the student (relationship? assigns? unique?)
Here's my personal best attempt, in 3 parts. Each part of the definition would include a discussion and examples before moving to the next part.

A relation is a set of links between
  two sets.
Each link of a relation has an input
  (in the starting set) and an output
  (in the ending set).
A function is a relation where every
  input has one and only one possible
  output.

I'm somewhat happier here: starting with a relation gives some natural examples and makes it easier to impart the special importance of a function (which is "better behaved" than a relation in practical circusmtances).
But I'm also still uneasy ("links"? A set between sets?) and I was wanting to see if anyone had a better solution.

REPLY [2 votes]: All of the above state the usual metaphors with more or less flair; but none convey the idea of a function. Functions have nothing inherently to do with machines or monkeys or rules (whatever they are) or black boxes (whatever they are) or inputs and outputs or the equals sign. There are reasons why we want the invention of a function. There are reasons why we need the invention of a variable. There is a reason why we need the notion of dependence that leads to the idea of an "expression" to provide what we want. The expression is a brilliant device; and it's a shame not to see that explicitly. There are reasons why we consider a constant to be a special case of a function. There are reasons why we commonly limit the word "function" to single valued functions. There are reasons why we expand the notion of a function to a function equation with an explicit dependent variable. And like other objects in mathematics, the same function can have different forms, algebraic and otherwise. Stating conclusions without their sense makes mathematics a mystique instead of a rational interesting science that we can think about and question and explore. One last point: The question should have been how to explain, not "define." No so-called "definition" comes out of nowhere. Presenting a "definition" like Athena springing full grown and armored from the head of Zeus is, to the great misfortune of most of us, standard practice in teaching mathematics.