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 May 12th, 2010, 04:23 PM #1 Newbie   Joined: May 2010 Posts: 6 Thanks: 0 Help on an Analysis Question 1. Let g(x) be a fcn that is continuous on its domain and de fined for all real x. Suppose that lim (x-> negative infinity) =infinity and lim (x-> infinity)= negative infinity. Is there is a value of x such that g(g(x)) = x.
 May 12th, 2010, 04:43 PM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Help on an Analysis Question Use the IVT on $f(x)-x$ to show that there is a point where $f(x)=x.$ It follows that at this point, $f(f(x))=x.$
 May 12th, 2010, 07:28 PM #3 Newbie   Joined: May 2010 Posts: 6 Thanks: 0 Re: Help on an Analysis Question Hi mattpi, Thanks so much for the reply. I was wondering if you could possibly elaborate. IVT on "F(x) minus x"? I'm familiar with the theorem but am having trouble seeing how I can apply it directly given the problem description
 May 13th, 2010, 03:18 AM #4 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Help on an Analysis Question You should be able to see from the description of $f$ that $f(x)-x\to\mp\infty$ as $x\to\pm\infty.$ If you look at what this means formally, you should be able to define a closed interval $[a,b]\subseteq\mathbb{R},\ a\,<\,b$ such that $f(a)\,>\,0$ and $f(b)\,<\,0.$ The IVT then tells you that there exists an $x\in[a,b]$ such that $f(x)-x=0.$
 May 13th, 2010, 07:26 AM #5 Newbie   Joined: May 2010 Posts: 6 Thanks: 0 Re: Help on an Analysis Question Ok, great. I can actually see that now, Bolzano's Theorem. Nice! How does that imply that f(f(x))=x?
 May 13th, 2010, 09:03 AM #6 Newbie   Joined: May 2010 Posts: 6 Thanks: 0 Re: Help on an Analysis Question NVM, I got it. Thanks!

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