# In Need of some brilliant ideas regarding ODD FUNCTIONS!!

Question1. Let $F_n(x)= \left( f_1 \circ f_2 \circ \ldots \circ f_n \right) (x)$ where $n \geq 2$. Find all (kinds of) functions $f_1,f_2, \ldots$ such that for all $x$ in some (nice) interval, $F_n(-x)=-F_n(x)$(we will now call this odd function) where $f_i(x) (i=1,2, \ldots, n)$ is continuous and not an odd function? (For the sake of sanity, let us exclude cases like the following: $f_1(x)=g(x+c), f_2(x)=x-c$ where $g$ is some odd function and $c$ is constant. I hope we are on the same page here about the interesting cases. Can you find other interesting examples? Maybe some mathematical ingenuity and creativity comes into play?)

Here are some well-known examples for $n=2$:

(Note: Let us exclude the case where we define $f_2$ as the inverse function of $f_1$ or vice versa, ie $f_1(x)=\ln(x), f_2(x)=e^x$)

Case1: $f_1$ being a logarithmic function and $f_2(-x)f_2(x)=1$ so that we can use the identity $\ln{\frac1x}=-\ln(x)$ $f_1(x)=\ln(x), f_2(x)=\frac{1+x}{1-x} (-1

$f_1(x)=\ln(x), f_2(x)=\sqrt{x^2+1}+x$

$f_1(x)=\ln(x), f_2(x)= \frac{1+ \tan(x+\frac{\pi}8)}{\sqrt2} (-\frac{3π}8 (in fact, the third example is the only example I can find which involves two transcendal functions.)

Case2: (Any peculiarities?) $f_1(x)=\frac{x-1}{x+1}, f_2(x)=e^x$

Question2. T/F

(1) Define $G_n(x)$ as the n-th iterate of $g$. $\forall x, G_n(-x)=-G_n(x) \iff \forall x, g(-x)=-g(x)$ (-> FALSE STATEMENT)

(2) There always exists some $g$ for any odd function $G_n$, where $g$ is not an odd function.

Question3. Using the function $G_n(x)$ from Q2, find a solution $g$ such that for all n, $G_n(x)$ is not odd BUT $\lim \limits_{n\to \infty} G_n(x)$ is odd, where $g$ is not odd.

LEAVE A NOTE FOR THE FOLLOWING: Errors, better notations, more examples, anything helpful, etc

Note by Inquisitor Math
2 months ago

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For question 1, I'm not sure how you might define "kinds of composition". You can get a boring (but infinite!) set of examples by using functions that are translations of odd functions, then applying a translation as the final function of the composition. For example, $f_1(x)=(x-1)^3$, $f_2(x)=x+1$. Neither of these is odd, but their composition is. Similarly you can combine arbitrarily many linear functions.

For q2, I think induction might work: $F_1(x)=f(x)$ is odd; then assuming $F_n(-x)=-F_n(x)$, $F_{n+1} (-x) = f(F_n (-x)) = f(-F_n(x))=-F_{n+1}(x)$

A question back to you, though: is it possible to compose a non-odd function $f$ with itself infinitely many times and get an odd function as a result? (I've only just thought of this so it may be trivial!)

- 2 months ago

Can you think of any solutions for $n=2$ other than the example I posted? Thinking of a solution for this is challenging, let alone the general case.

- 2 months ago

Ah, sorry, I think I misread your q2. Well, the proof above works one way, at least.

Here's another $n=2$ example; again it's part of a larger family, but, again, not a very interesting one... $f_1(x)=1-\frac{1}{x},\;\;\;f_2(x)=\frac{1}{1+x}$

Actually, you also get $f_1(f_1(f_1(x)))= x$

- 2 months ago

$f(x)=\sqrt{2+x}$ is non-odd and non-even. When composed with itself infinitely, it gives $F(x)=2$ which is even. Can a trick like this be used to find an odd function?

- 1 month, 1 week ago

Seems that you are interested in this discussion!(Particularly question2, yes?) Well honestly, I haven’t thought about infinite compositions that approach some odd function. But here is a function that works out (kinda disappointing that it doesn’t approach something.. if you seek of a general solution I dunno.), assuming that we are focusing on Q2: $g(x) = \begin{cases} 0 & \quad (0 \leq x \leq1)\\ 1 & \quad (x<0 \text{or} 1

(in fact, $G_n(x)=0$ for $n \geq 2$)

- 1 month, 1 week ago

Yes, I'm interested in whether there's a non-odd function that approaches an odd function after infinite compositions with itself. I'm not any expert and have just scratched the surface of this so far. Your function $g$ above actually fits the bill. It feels wrong though, perhaps because a function that is zero everywhere (so is technically odd) seems trivial somehow. I have no idea how to approach Q2(2). I feel like there should be some trick to allow it to happen but it depends what is allowed. It is possible if the function can vary with $n$ e.g. $(x+1-n (\mod 3))$

- 1 month, 1 week ago

I think that the question I am asking is this: Is there a function (real to real) that does not vary with n, is not odd, is continuous over the real numbers and when composed with itself infinitely many times approaches a non-zero odd function as a limit?

- 1 month, 1 week ago