Saturday, July 19, 2008

What's the Function of a Function?

While watching a 53 year old golfer winning the Brittish Open I decided to tackle the discussion on functions. I feel pretty comfortable with how to teach about functions. What has me struggling a little bit is the "why". I understand the value of functions, especially as they relate to Physics. As well, I understand the importance of one-to-one and many-to-one concepts in Abstract Algebra and the importance of domain and range. But my question is:

Why is it important to teach an Algebra 1 student if a relation is a function or not?

Any takers?


Patrick Regan said...


I think that one use of functions is the limitation placed on the mapping of the domain to the co domain. Namely an element of the domain can only map to one element of the co domain. What this means is that you can't get more than one output for the same input. Imagine if it were possible to have an element of the domain map to more than one element of the co domain. Basically, in this case, you would not be certain what you were getting out of the function. One way to think of functions is as a black box or a machine. You put a number into the black box/machine and another number comes out. Imagine if you had a machine or electronic device that gave you different answers for the same input. For example, say your steering wheel sometimes turned left when you rotated it right but sometimes turned right when you rotated it right. Your steering wheel would be useless. In the same way a function would be useless if it did not do the same thing each time. So I think the usefulness of the function idea is to set up a guarantee that things will turn out the same way each time you put the same number into the function. That is the only thing I can think of.


Trent Tormoehlen said...

I don't disagree with that. One-to-one functions and many-to-one functions are very interesting to study. But they are a very theoretical part of Algebra.

I think the discussion is much bigger than this. What is the goal of the math classes we teach in High School? I have some ideas of my own to that question, but I know that I don't have the experience to have a fully developed answer.