Summary: A Grammar for Functions

How is it that we are able to see or hear a sentence in our native language and, from among all of the infinite possible sentences that might be constructed, recognize the sentence as both grammatical and meaningful in a particular situation? Certainly we can not "know" beforehand all of the infinite possibile "correct" combinations of words. Instead, what we know is a basic vocabulary and a grammar. A grammar is a method for generating and parsing "correct", useful sentences from a limited vocabulary.

The situation is very similar with our study of functions. We can not hope to "know" them all. We can not keep, on the tips of our tounges, descriptions of every possible input/output situation that may someday, somehow arise in application. Instead, we learn a few basic functions and a grammar for constructing new ones. The methods of this lesson show you some of the basic components of a functional grammar.

As in everyday language, some people have bigger vocabularies than others. We assume that most everyone knows some basic words/functions, but individuals develop specialized extended vocabularies that are based on their experiences.

Don't fret if you feel that your functional vocabulary is currently quite limited. You'll learn about many new functions in calculus. More importantly, you will construct many new functions in calculus, using the basic grammatical methods covered in this lesson.

It isn't those with the largest vocabularies that succeed in mathematics and its applications. It's those who can think on their feet, and construct a clever turn of phrase when necessary.