Introduction: Given  f ...

Given  f...

The phrase is one of the most common introductions in mathematics. It means something like this: "Hello. I assume you're acquainted with  f . Let me tell you something you don't know..."

The introduction carries with it an unspoken supposition that the function  f  is already "known" to us — somehow. What usually follows is a description of something that you can then do to  f  to arrive at a brand new function, one that you may not "know" as well, but one that is better suited than  f  itself for solving a particular problem.

These kinds of "conversations" — about how to build new functions from old ones — occur every time mathematics is used to solve specific applied problems.

Why must we constantly build new functions?

Part of the reason is that we can only be expected to "know" a small subcollection of the infinite number of input-output rules that functions describe. In precalculus, we spend a great deal of time becoming acquainted with a few common families. In calculus, we learn more about these families, focusing on how the special capabilities of individual members can be applied.

No matter how hard we study, however, it is impossible to become familiar with all of the possible families, and all of the individual members of those families. The world of functions is too vast.

Even in specialized fields, we can't anticipate all of the cause-and-effect relationships that might occur in applications. Ask a physicist about the functions that she is acquainted with and you may get a list of interesting names: delta functions, Bessel functions, Green's functions... Talk to that same physicist about the specialized functions used in engineering, biology, or psychology, however, and she may just scratch her head and shrug her shoulders.

Which functions, then, should you "know"?

Everyone should become acquainted with some basic families of functions. That is exactly what you are doing in precalculus.

Mathematical experience, however, teaches us that we must also learn to be very handy with these functions. It is a "handiness" with building new functions, much more than the basic toolkit of functions that we begin with, that gives us the resources to build mathematical models as we need them.

In this lesson we'll begin to learn some of the methods that are used to construct new functions from "known" ones. After you've mastered a few simple procedures, you'll be able to nail together familiar functions into all sorts of useful shapes and sizes.