This section introduces the idea of studying universal properties to avoid memorizing vast amounts of information.
As we have mentioned, functions are fairly general things; just a relation with a domain and codomain, and the additional (very important) feature that each element of the domain is mapped to a single element of the codomain. There are a number of different types of functions that will be useful to know (like those listed in the ‘library of functions’), but before we do so, it is helpful to learn about the kinds of things we can do to all functions.In an effort to devalue memorizing, and promote learning, this topic aims to teach a number of properties and techniques that are universal to all functions, so that you don’t have to memorize/learn properties and techniques for each function type, thus hopefully saving time (and sanity) for all involved. Moreover we will learn what kinds of things are of universal interest to know or study regardless of function type, which will (hopefully) motivate some of our future function exploration.
There are a number of ‘classic’ function manipulations that have very clear and concise graphical results, and we study those next. Generally these manipulations fall into two broad categories; rigid translations, and transformations.