So I've been going through some of my textbooks to keep my mathematics knowledge up on point, and I can across this problem which I'm not quite sure how to solve. I've tried a few methods (I'll put my attempts below), but I just can't find a good way to solve this.
Let's define $f(x) = \sqrt {4x^2-4x^4}$. Consider the equation $f(f(f(…(f(x))…))) = x$, where there are a total of $2020$ functions $f$ being applied to $x$. And let's say a total of $A$ real numbers satisfy this equation. Find the value of of $A$ (Mod $1000$).
I first though of trying to take the amount of functions that it would take to get a certain number back to itself, and check whether that would be divisible by $2020$, because if it is, then that means it will satisfy the condition. I tried to do this for a while, but I just couldn't make sure I was $100%$ accurate and I wasn't sure when to end my search either. And considering the problem wants me to give the answer mod $1000$, I figured that brute forcing it like this would take too long. So then I tried to look for a pattern by looking at the first few numbers that would work, but I just couldn't find any detectable pattern or similar properties of those numbers. And that brings me here, where I am asking for some help to find an efficient method for this problem or possibly a formula for this kind of problem. (And I want to learn from this so a decent explanation or guide would be nice)
Okay so I've got an answer of 576 from somewhat brute forcing this (using above method), but I really want to see if anyone can spot a pattern and confirm this, or use a formula to get an answer and explain their logic.
Best Answer
$\mathcal { Hint:} $
$$x=\sin(t)$$ $$f(x)=\sin(2t)$$
After $k$ iterations
$$f(f(...(f(x))...))=\sin(2^kt)$$ Also, since $x \geq0$ , we need to check solutions in only $0 \leq t\lt \pi$.
See if this helps.