Heh. I make simple mistakes like that a lot. It's so annoying when you can do something tough and then you make a mistake like as a smaller part of the process.

I'm not sure where you're getting f(x) = x^2 - 4 from. The function should just be x^2. The output of the function is 4. 

When you draw the inverse of a function, you have to restrict it to a one-to-one function before doing so. If the function is not one-to-one (which the quadratic function isn't), then it does not have an inverse.

I'm not taking the inverse of a function. 

x^2 = 4 subtract 4 from both sides to make x^2 - 4 = 0 then you can evaluate it at y=0.

Either way, -2 and 2 are solutions to both functions. f(x) = x^2, where f(x) = 4. This is just shifting the graph to evaluate at a different y - value.

It's not if you started from x=2.

That's a variable declaration, I'm done talking about this.

There is no function.

Certainly f(x) = x is not the same as f(x) = x^2, but this isn't a function we're analyzing, so I'm tired of discussing it. It's obsolete.

Thread is huge.

Well I think an easy way to visualize it is that 0.9999.... represents the end of the 0. ..... cycle and 1 represents the beginning of the next 1. ..... cycle.

The exact end of a cycle and the beginning of the cycle are the same, e.g  2400hrs and 0000hrs are the same. The only way to represent the end of the 0. ... like the 2400hr case is  0.9999..., there is no other way to write it.

Just think about it as 0.9999... means that all the decimal places for the 0. .... cycle are filled ( hence the infinite 9's, and adding a 0.000....001  anywhere will give you something greater than 1). So it's the end of the 0.... cycle, and the beginning of the 1. ..... cycle. It's the same thing.

Edit: oops I think I made it sound more complicated..... ah well

All maths problems effectively start from a "variable declaration".

Like, 3x-5=0 is equivalent to a declaration that x=5/3

But if you can't see that straight away you can do a number of steps. But really everything single step is the same as saying "1=1".

Ironically, a major stumbling block for beginners in computer science is that they have a muddled mental model of variable assignment.  When they see an assignment statement like "x=2", a part of them confuses this with the equals sign in math.  In mathematical writing, "x=2" is not a variable declaration but merely states that the two sides are equal (one declares a variable by using words like "let", "define", or sparingly, the shorthand ":=" symbol).  So the sentence "x=2" holds no more or less information than "2=x".

[One exception to this claim: in my own area of math, we often (ab)use the = sign in a way that is not symmetric when describing error estimates.  We would write "x = O(n)" to mean that x is bounded by a multiple of n, when formally it would be more appropriate to use element inclusion.  Consequently, even if one has "x = O(n)" and "y = O(n)" one cannot deduce that "x=y".]

By contrast, in (declarative) programming languages it makes perfect sense to write "x=2" followed later by "x=3", which is mindboggling to a certain number of beginning students.  And of course in most languages "2=x" would be an illegal assignment.

That's why the notation x=O(n) is not so efficient and shouldn't be used. You should say that x= n h(n) with h(n) tending to a constant when n is getting large.

@dsgrue3: I have a good example for you, maybe it will be a good clue to understand why you cannot consider -2 as a solution even if it solves your equation x²=4.

We start from the equation x=2. Then we notice that if x is actually equal to 2, then x>0. But 3 is also greater than 0. Does it imply that 3 is equal to 0?

Of course not, because in the second step we have "weakened" the hypothesis by switching to an inequality, but we obviously didn't change the fact that x is the solution to the equation x=2. In your own case, you also "weakened" your first equation by taking another equation x²=4 which doesn't exactly replace you equation x=2.

This is just an example that you cannot "easily" replace a mathematical object by another, even if they look similar. Just like you cannot replace the function f(x)=x by the function f(x)=x²/x which is not defined for x=0. 

No, that's meaningless. All I did was abuse substitution as the OP did. 

10x = 9.999...

Could lead to 9x = 9 or by substituting on both sides 9*0.999... = 9 (which of course is true only if 0.999... = 1, which is what we're trying to prove. It's circular logic.)

I don't disagree, but I don't see a point here anywhere. Given x = 2 or 2 = x, there is nothing to solve.

No, they END with a variable declaration. You're solving for x.

You don't start with a declaration and then manipulate it, there isn't a point as it's already known.