n00b1sh math notes, with sean.

Adv. Pre-calculus, section 1.4

I was trying to think of a way to make these [very short] notes fun, and then Shan-Wen/Sean popped into my head and was like ">:D Use meee!" And I love him, so I obliged. Hopefully I can use this to channel my inner asian hipster.

Eyyy bros! What up in da hizz-
Okay, really, Sean? There are notes to be taken. -nods at textbook-
Right, sorry bruh. Here we go. Thus far, I/we have learned advanced algebra concepts, as well as set and relation theory. I/we can plot points on a grid and know the different sets of numbers.
Our objectives in this chapter are to find stuff's domain and range, figure out if it's a function and junk, and figure if two functions are equal. Chyah.
The section starts out being all like 'blah blah, cipher blah blah.' Essentially, it's an unimportant load of shiz that the teacher's not gonna get up in my grills about. Plus, they left some cipher and the mofo who had this 20-year-old book before me tried and failed to decode it. Pff. Sean, you digress.
Right, right. So anyway, what is important is that they say when you got two corresponding sets and stuff, you "map" them, and it's called a relation. This, of course, is what our math class has spent the last three weeks learning.
So we already know that a relation is a set of ordered pairs, and the first ones in the pairs make a set called the domain, and the second ones in the pairs make a set called the range. Apparently, a function is just a special type of relation.
A function "f" is a relation in which each element in the domain is mapped to exactly one element in the range. The expression f(x) refers to the value of f at x.
Chyah bro. Anyhow, the range is represented by R_f and the domain is D_f. Plus, functions are rules, yo. So stuff up in the function's gotta adhere to its rules and whatnot.
If you're plottin' a function and junk, and it don't specify what the domain is, bro you just gotta assume it's the real numbers. Only real numbers that can't be in the domain are ones that give a denominator of 0 or the even root of a negative, yo. And if you dunno what that range is, you can put it all up on your graphing paper.
So basically after all that junk, the textbook's like 'yo man, here some practice problems for yo brainnn' but I dunno, I just am not feelin that. Plus I kick you-know-what in maths class so it's like, man whatever, I don't need yo freakin practice problems and crap.
...

Right, right, no worries. We're pretty much like halfway or something, I dunno could be three quarters. It's chill, yo.
...
Okay, so basically, f(x) is pretty much the same as y, given that your function is in terms of ordered pairs (x,y).
Two functions f and g are equal functions if the following two conditions are satisfied:
The domain of f is equal to the domain of g
For every x all up in the domain, f(x)=g(x)
For you or me to know that a relation's actually a function and junk, you gotta know that with (a,b) and (a,c) b and c are equal. Otherwise it's not legit. It's a poser function.
Whoa! Now the book gets all up in some random shiz.
If you gotta function graph, and it keeps going and stuff, i'ts continuous or "unbroken." If it's got a gap at a point all up in its line, you know you got a discontinuous graph.
But we'll talk about that stuff later. [Chapter 16, to be exact ^^]

K, so if you're super noobsauce and junk (like me) then you don't have to make yo relation like a set or map or rule; you can just graph that sucka and then put a vertical line all up in it. Again, this only works if you're noob. (No, seriously. I actually use this all the time...)
So if you've got a function and stuff, it's one-to-one if for 2 elements all up in its domain, f(x₁) = f(x₂).
Meaning that if you have two different x values go to one y [like in a parabola, which is a U shape] it isn't a one-to-one. A one-to-one is a function such that each element in the domain plots to one in the range and vise versa, thus the name.
You can do the horizontal line test for this. But it's pretty simple and stuff so yeah.

Yea bro, that's all! Good thing that was only 4 pages, yo...
Yes, Sean, you did wonderfully. Now get back in my skull. Can't have you bouncing all over the place.

:D

Adv. Pre-calculus, section 1.6

I didn't feel like taking notes on 1.5 because it was stupid and easy and junk. It was just doing operations with functions, so back to basic algebra. f(g(x)) yada yada. I pwn at that stuff.

Anyhow, onto our topic o' the day...
Inverses!

So you've got one'f those relations. [Ordered pairs and all that, you know the drill homie.] You switch the x and y of those coordinates and bah-dum, you still gots a relation! But you try'n do the same thing with a function - you switch the x and y of it's coordinates and junk - and you won't necessarily end up with another function. So like, homie, take this:

{(2,4),(-2,4),(3,9),(-3,9)} <---TOTES A FUNCTION!
and its inverse:
{(4,2),(4,-2),(9,3),(9,-3)} <---TOTES NOT A FUNCTION D8

'Cause see? Like, all up in that first function, each element of x only goes to one element of y. But in the second one, you got elements of x pullin' crazy stunts and gettin with multiple y's if you know what I'm sayin'!
...Really, Sean?
Sorry bruh. Anyhow, the book highlighted a definition so it's probs important:
A function f has an inverse function g if and only if f(g(x))=x for all x in the domain of g(x) and g(f(x))=x for all x in the domain of f(x).
...? D: -does not understand-
No worries, Creator! All that means is that you've got two functions, right? Right. Mkay, so if you're wondering whether two given functions are inverses, you just see if the domain of one is the range of the other. Okay, that's all well and good, but I still don't get that weirdo definition the book gave. Think of it like this: g(x) is the range when x is the domain, right? Yep. Okay, so think of that g(x) as its own single variable, say w. Then, if you took a function f(w), that's the range and w is the domain, right? Ohhh! I get it! Thanks Sean :D Your azn math skillz are quite useful, you know.
Dude, anytime! BTW, the book wants us to note that the inverse is denoted all like f^-1 or f^-1(x) . It also wants us to make sure we knows and stuff that that's all different from a reciprocal, which is (1/f(x)) and denoted [f(x)]⁼¹ . Also, you don't really need to know that, it's just the textbook being a craxydude and junk.
...K? Anything else I should know?
If you gotta find an inverse or whatevs, and it's all up in the equation form, you just switch the x and y and solve for y.
Oh, good, something I remember from Algebra II! Cool beans, anything else in the section?
Well, the obvious - any relation's inverse on a graph is that relation reflected across the line y=x.
Duh.
Oh, and one last thing brosky! If a function's got an inverse that's also a function, it's gotta be a one-to-one!
OHHH. That clears up why I was confused earlier, 'cause I thought it was trying to explain how to tell if a function has an inverse function but it was actually just showing if two functions are inverses.
Chyah.
Thanks so much, Sean! See, this is why I love you. You have ridiculously pwnage skills in math which are far better than mine...?!?!
No probs bruh! I'll seeya later since you'll undoubtedly require more assistance in the future.
:D