Functions

You may be unable to name the highest peaks of the Moluccas — remote islands in the Malay Archipelago — and yet still be quite familiar with the many spices that grow there. You can stir nutmeg into holiday eggnog, and appreciate the flavor, without the slightest knowledge that the nutmeg's 400 species of tropical evergreens cover the Moluccas' spectacular mountains and volcanoes.

Your experience with functions is perhaps somewhat similar. You've probably heard of functions. You've probably tasted their distinctive mathematical "flavor". You may even recognize some of their many varieties. And yet, faced with the information that there are far more than 400 types of function, covering the mountainsides in the foreign land of Calculus, you may begin to doubt if you know them at all.

Precalculus will talk about functions in a way that may, at first, be unfamiliar to you. It would be a mistake, however, to tell yourself that you really don't know functions at all. You probably know a lot about functions!

For example, you've probably worked with equations like this:

y = x² + 2x + 3 ,

drawn graphs in an x,y-coordinate system like this:

and filled in tables of values while collecting experimental data, like this:

These are all functions! They may look like quite different things, but they can all be given a very general, unifying description.

Here's how.

Each of the examples above involves a pair of values: x  and  y . Lots of things come in matched pairs like this.

Functions are just the rules for finding the one thing that uniquely corresponds with another. In the examples, there is always a unique  y  corresponding to each  x . The rule tells you how to find the proper mate: plug into the equation, read the graph, read the table.

If you can appreciate how each of these examples fits this very general notion of "function", then you've already made your first steps on the bridge towards Calculus.