1. ## Two questions about function composition

#1
I have a continuous, one to one function f(x). How do I find g(x) such that g(g(x)) = f(x)?

#2
Also, if i forever keep feeding the output of a continuous one to one funtion back into the input, isn't the result also continuous and one to one?

{i mean ...f(f(f(f(...f(x)))...}

How do find the values of x for which this converges to some finite value?

Can I find what the value is?

thanks

2. Originally Posted by kd8bxz
#1
I have a continuous, one to one function f(x). How do I find g(x) such that g(g(x)) = f(x)?

#2
Also, if i forever keep feeding the output of a continuous one to one funtion back into the input, isn't the result also continuous and one to one?

{i mean ...f(f(f(f(...f(x)))...}

How do find the values of x for which this converges to some finite value?

Can I find what the value is?

thanks
Read this: Iterated function - Wikipedia, the free encyclopedia

3. Originally Posted by kd8bxz
#1
I have a continuous, one to one function f(x). How do I find g(x) such that g(g(x)) = f(x)?
Sorry, but such a function g (with the same domain and range as f) need not exist. Consider for example, the function $\displaystyle f: M\ni x\mapsto -x\in M$, where $\displaystyle M := \{-1;+1\}$. In this example it is easy to have an overview of all the possible Functions $\displaystyle g:M\rightarrow M$. Do you see one that satisfies $\displaystyle g(g(x))=f(x)$ for all $\displaystyle x\in M$?

#2
Also, if i forever keep feeding the output of a continuous one to one funtion back into the input, isn't the result also continuous and one to one?

{i mean ...f(f(f(f(...f(x)))...}

How do find the values of x for which this converges to some finite value?

Can I find what the value is?

thanks
Well, if f is continuous and if the limit $\displaystyle \lim_{n\to \infty}f^{n}(x)$ exists, it follows that such an x must be a fixed point of f: that is, x must satisfy the equation $\displaystyle f(x)=x$. In other words: to find those xs you solve the equation $\displaystyle f(x)=x$.