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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.

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    Most answers given below will suggest if not outright require the function to be computable. This rules out constructs like the busy beaver function. It might be that for the audience you have in mind, this kind of white lie is quite acceptable. But your original attempt does not have this drawback, so perhaps someone can come up with something that is easier to read / understand / imagine and yet doesn't require or imply computability.2013-07-30

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