[Coda]

Originally Posted by Potironette

Or maybe when a function gets inversed(?) people like them to remain functions and so limit them..? That doesn't really seem it though.

Actually, that IS it. For the same reason, square root is constrained to only the POSITIVE square root of the number, even though its inverse operation, x^2, exists for all x.

(if I just plot x = sin(y) or x = cos(y) I get a relation on desmos)

I'm not sure what you mean by a relation in this context. Usually when I think of relations I'm thinking of sets.

On the other hand, arctan can go on forever in the x direction and still give one output/be a function which is why they get to go on forever?

Yes. Like I said, it's because tangent goes to infinity in both directions between -pi/2 and +pi/2.

The bit about the video games was really interesting :o. Although, though I have a sense that angles are useful in video games I don't really know why figuring out an angle based on a coordinate would be useful.

You locate objects in space by their coordinates. A simple example would be a 2D game that counts pixels as the coordinate system. If you have a bird enemy that can dive-bomb the hero, then it might be flying along at y=600 while the hero is running on the ground at y=0, and when it gets close enough to see, it needs to figure out what angle to dive at in order to attack.

I found this in my review sheet but I have no clue what it means:
Regular division: p = qd + r, r < d (dividend = quotient x divisor + remainder)
p(x) = q(x)d(x) + r(x) deg(r) < deg(d)

Taking d(x) = x - x_0,
p(x_0) = 0 <-> r(x_0) = 0 <-> p(x) = (x - x_0)q(x), i.e., (x - x_0)|p(x)

(It's the "Taking" part I don't understand. I'm pretty sure deg(r) < deg(d) is something about how remainders don't have the same "degree" as divisors(?), though I don't really know what a "degree" is. I used "<->" to replace the arrows in my review sheet. I'm not sure what "|" is supposed to be, though I vaguely remember reading somewhere it means "divides")

I had to look this up because I couldn't figure out the context: This is specifically talking about polynomial division.

The degree of a polynomial is the highest exponent in it -- the degree of "x + 1" is 1, the degree of "x^2 - x + 1" is 2, the degree of "x^3" is 3, etc.

The degree of the remainder in a polynomial division MUST be less than the degree of the divisor. Think about it by analogy with regular division -- regular division is like repeated subtraction, and if the remainder is larger than the divisor, then your quotient isn't big enough and there's still room to subtract more divisors from the dividend.

The double-headed arrow means "if and only if", and it means that if the left side is true then the right side is also true, and if the right side is true then the left side is also true. For example, "2x = x <-> x = 0" says "if 2x = x, then x = 0, and if x = 0, then 2x = x". On the other hand, "x = 2 <-> x^2 = 4" is invalid, because while "if x = 2 then x^2 = 4" is true, "if x^2 = 4 then x = 2" is false because x could also be -2. (You would just write "x = 2 -> x^2 = 4"; the single arrow is read "implies".)

I'm not super familiar with the property as it's written, but some searching around shows me that this is a property of a polynomial zero. From there, I understand a lot more about it.

"Taking" there could also be interpreted as "suppose" -- that is, suppose you have d(x) = x - x_0. The following must either all be true, or all be false:

1. p(x_0) = 0 -- that is, the original polynomial crosses the x axis at x_0, or equivalently, x_0 is a root of p(x)
2. r(x_0) = 0 -- that is, the remainder of p(x) / (x - x_0) is 0 at x_0
3. There exists some q(x) such that (x - x_0) * q(x) = p(x) -- equivalently, (x - x_0) is a factor of p(x).

This is a formal way of saying "(x - x_0) is a factor of p(x), if and only if p(x_0) = 0".

For example: We can show that the function (x^2 - x) = 0 at x = 1. This property lets us immediately know that (x^2 - x) is divisible by (x - 1). It doesn't tell us what the result of the division is, but we know that a result exists. But it equivalently means that if we know a degree-1 factor of a polynomial, we know that it must touch the x axis at that point.

EDIT: Below those equations on dividing, the Remainder Theorem, Factor Theorem, Fundamental Theorem of Arithmetic, Fundamental Theorem of Algebra, and Rational Root Theorem are listed out...and they all refer to p(x) = (x - x_0)q(x). I guess it means to say that somehow, x - x_0 got rid of the remainder, somehow.?

It means that all of those theorems only hold if p(x) is evenly divisible by (x - x_0). If it's NOT evenly divisible (that is, if r(x) isn't zero) then the theorems don't apply.