before starting a journey, it's important to know why we're going on it. for starters, why analyze functions? it might seem trivial to ask, but functions are the meat of how we describe the world around us; every interaction we've ever had, everything we touch, our world as we know it is comprised of relationships between things, which is all that a function is. and in order to explore the world and then make things in that world, we're going to need to understand these functions.

in this book, we're going to define a map as an object that takes us from one space to another space, the most general possible term for these kinds of object. a function is a map which goes from any space into R or C. any dimension version of those spaces is alright, as long as it goes into those guys. i'll also do my best to be consistent about using "linear map" for maps that are actually linear. there might be times where i call a function a map, and many times which i'll just say function. the two are often used interchangably, or differently by other authors, so keep this in mind!

to quickly recap functions, we're going to want to think of them as an object which takes an input and produces an output- kind of similar to the abstractness of computer science functions. if we have a smooth function, that means the function can be drawn evenly with a pencil in one line- no holes, and also no sharp corners (we'll learn that those sharp corners- cusps- create some trickiness for some operations we'd like to be able to perform). functions can be real-valued or vector-valued, or really anything-valued, it just depends on what we're putting in and what we're getting out. hopefully you feel comfortable visualizing functions in R2 and working with them, including exponential, trig, and other special guys! if not, you'll get to know a very different side of them through this book.



so how do we get to know them? well, we've already had a bit of a taste of that from our foundational math skills, we can look at a picture to see what a function is, get values from it, and so on. but now we want to categorize these functions. we want to know their whole life stories- where they've been, what they've done, and who they're going to be. we can classify functions in tons of different ways, some of which we'll see later on in this book, but for now, we're interested in what i call the practice of divination!

in order to describe anything fully, we need to know its past, present, and future. we've already got the present: put an input into a function, get an output. but what about the past and future? divination is the art we employ to get that information about a function, through differentials and integrals. those words don't mean anything to you yet, but they're wyld and exciting- we're going to be constructing our own spells that allow us to divine that info just by knowing a single point and the function itself!

now, divination isn't easy, i'm sure you can see that. we've kind of got no leg to stand on right now. so we'll dive into some tools we'll need to wield in order to create these spells.