Supermum of $f(x)=\frac{x}{\sqrt{1+x^2}}$ composed $n$ times on itself

Question

For a natural $n$ and $f(x)=\frac{x}{\sqrt{1+x^2}}$, find the supremum of the function:

$f^n(x)=f\circ f\circ...\circ f(x)$, when $f$ is composed $n$ times on itself.

Answer 1

You are probably interested in a general approach to such problems. There is none. Forget it. In rare cases you might be able to find the explicit form of $f\circ f\circ...\circ f(x)$ and everything that follows. This is our case. See: $$f\circ f(x)=\frac{x/\sqrt{1+x^2}}{\sqrt{1+(x/\sqrt{1+x^2})^2}}=\frac{x}{\sqrt{1+2x^2}}$$ Now $$f\circ f\circ f(x)=f(f\circ f(x))={x/\sqrt{1+2x^2}\over\sqrt{1+x^2/(1+2x^2)}}={x\over\sqrt{1+3x^2}}$$ Now you have enough material to guess the general expression for $f^n(x)$, prove it by induction, and finally get the supremum you were after.

Answer 2

Typically for problems like this, i.e. possibly homework problems, you should say something more than just the question. In particular, do let us know if this is homework, and also what you have tried or what you have thought about concerning the problem and the difficulties you run into.

Now, as for your problem, I suggest that you consider some really simple cases first. Like, painfully simple! So, let's only focus on $n=1$ for the moment. What can you say about the function $f(x)=\frac{x}{\sqrt{1+x^2}}$? In particular, because you are taking $\sup$'s, is there a portion of the real line you can automatically discount from consideration? For the rest of the reals, can you say something about the behavior of $f(x)$? Does it behave in a very nice, regular fashion, or does it wiggle uncontrollably? What happens when $x$ gets super large?

Once you have answered those questions, you should have a good idea of what $x$ must look like for you to be approaching the $\sup$ for $n=1$. But now when you iterate the composition, all of your analysis will remain the same!