I teach a college algebra class, which is designed as a service course for most other disciplines at our University. One of the first topics we cover is a short unit on functions. Even though these students are supposed to know things like what constitutes a function, what the domain of a function means, etc., most of my students react as if they have never heard the word "domain". So I begin the lesson with a short activity. I give them the definition of a function (if A and B are nonempty sets, a function is a rule that assigns to each element of A exactly one element of B). Then I set up an example using the people in the classroom as the set A, the letters of the alphabet as the set B, and the "rule" being that each person in the classroom is assigned the first letter of his or her last name. Every person then tells me his or her letter and I write it on the board. I identify the set of people as the "domain", the set of letters as the "codomain", and the set of letters I have written on the board as the "range". We make the points that:
1. Every person participated (so every element in the domain is assigned something).
2. No person participated more than once (so no element in the domain was assigned more than one element in the codomain).
3. Some letters were used more than once (so elements in the codomain can be repeated).
4. Some letters were not used (so elements in the codomain may not be used, meaning that there is a difference in the "codomain" and the "range".
We then go on to functions given as algebraic expressions, finding domains of functions, graphs of functions, the vertical line test, etc. My students seem to respond positively to this activity and the fundamental characteristics of a function are implanted without any references to "formulas" (which is what we deal with the most in our college algebra class).
Now, after that lengthy introduction, here's my question: Do any of you have any other quick, meaningful introductions to functions? I have been using this particular example for a number of years, and I would like to try something different. We do not discuss relations prior to learning about functions, so any discussions we have in our class must start from square one. The nature of our course prohibits any lengthy discussions of any of our topics.
[Non-text portions of this message have been removed]