Je bent je ingevulde velden bij deze pagina aan het verwijderen. Ben je zeker dat je dit wilt doen?
You are erasing your filled-in fields on this page. Are you sure that is what you want?
Nieuwe Versie BeschikbaarNew Version Available
Er is een update van deze pagina. Als je update naar de meest recente versie, verlies je mogelijk je huidige antwoorden voor deze pagina. Hoe wil je verdergaan ?
There is an updated version of this page. If you update to the most recent version, then your current progress on this page will be erased. Regardless, your record of completion will remain. How would you like to proceed?
This section introduces the idea of studying universal properties to avoid memorizing
vast amounts of information.
Functions: Why study ‘universal’ things and not just specific functions?
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 most precalc classes, many of the things
taught in this chapter are re-taught as ‘features’ of each of the function types. This
leads to students memorizing vast amounts of redundant information.) 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 (Technically some very advanced graduate level ‘functions’ don’t precisely
behave like the claims in this topic, but unless you aim to do graduate level
mathematics, you need not worry about this fact) 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.
Universal properties of functions: Graphic manipulation
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.
(There are
other function manipulations one can study, however other manipulations
become weird (like flattening graphs in some dimensions but not others, when
you are dealing with a function in 10+ dimensions). These manipulations
are (clearly) beyond the scope of this course and moreover even through
calculus almost all of the graphical manipulations one would like to do can
be re-written in terms of rigid translations and transformation, meaning
these two broad categories cover a surprisingly vast amount of applications.)
The virtue of studying universal properties in general is...
we don’t have to learn
specific things anymore.by learning a few universal properties, it cuts down
significantly on how many specific properties you must learn.universal properties
tend to be very important and useful properties that apply widely.we can get a lot
of the more annoying things out of the way early and never need to know them
again.
Precalculus Algebra MAC1140