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July 13th, 2017, 12:47 PM  #1 
Member Joined: Sep 2014 From: Morocco Posts: 36 Thanks: 0  Generalizing with a function
Hi, I'm given the function defined as follows, with c a real that's strictly positive. For each x in IR, $\displaystyle f(x) = \frac{x}{\sqrt{1+cx^{2}}}$. The exercise is to calculate the image of f(x), then the image of the former image, then lastly, to generalize. This is what I have managed to do, and I'd like to have an idea on how I can formalize a generalization. $\displaystyle f(f(x)) = f(x)\frac{1}{\sqrt{1+cf(x)^{2}}}$ $\displaystyle f(f(f(x))) = f(x)\frac{1}{\sqrt{1+cf(f(x))^{2}}}$ Thanks a lot. 
July 13th, 2017, 12:59 PM  #2  
Senior Member Joined: Aug 2012 Posts: 1,850 Thanks: 509  Quote:
On a notational note. IR is the worst of all possible display options for $\mathbb R$. You can see how I did that by quoting my post. Other alternatives are boldface R, or copy/paste from the excellent math.typeit.org. Or just say "the reals."  
July 16th, 2017, 12:16 PM  #3 
Member Joined: Sep 2014 From: Morocco Posts: 36 Thanks: 0 
Hi Maschke, thanks for the reply. I think what is required is to deduce a general formula based on those two results. I found this exercise under a text about induction. Thanks again. 
July 16th, 2017, 12:33 PM  #4 
Senior Member Joined: Oct 2009 Posts: 365 Thanks: 121  Correct, they want you to find a general formula. For that, you use your expressoin $f(f(x))$ and you substitute $f(x)$ in there with your function value, then you simplify. It's a bit of nasty algebra.

August 18th, 2017, 12:03 PM  #5 
Member Joined: Sep 2014 From: Morocco Posts: 36 Thanks: 0 
Hi again, sorry for the late reply. I simplified f(x) and f(f(x)) again, it turns out that I had to simplify more. What I then did was to conjecture the following: For each n in $\displaystyle \mathbb{N^*}$, (n iterations) $\displaystyle f \circ f \circ f \circ ... \circ f(x) = \frac{x}{\sqrt{1+ncx^2}}$ This was later demonstrated through induction. 
August 18th, 2017, 04:01 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,844 Thanks: 1566  
August 19th, 2017, 04:45 AM  #7 
Member Joined: Sep 2014 From: Morocco Posts: 36 Thanks: 0 
Hello, I found it to be: $\displaystyle \frac{x}{\sqrt{1+2cx^2}}$ 
August 19th, 2017, 08:01 AM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,099 Thanks: 850 
That is not what "image of a function" normally means.

August 20th, 2017, 09:11 AM  #9 
Member Joined: Sep 2014 From: Morocco Posts: 36 Thanks: 0 
I see, it must be some misunderstanding due to the differences between the English and the French languages. What I meant was f(f(x)).

August 20th, 2017, 10:34 AM  #10  
Senior Member Joined: May 2016 From: USA Posts: 996 Thanks: 409  Quote:
I am going to alter the notation a bit. $f^0(x) = x.$ $f^n(x) = \dfrac{f^{(n1)}(x)}{\sqrt{1 + c * \{f^{(n1)}(x)\}^2}} \text { if } n \ge 1.$ That is ONE generalization. But maybe they want a closed form in terms of c, n, and x. Finding that may be possible, but a bunch of algebra will be required. If that is what is wanted, I suggest that you calculate $f^1(x),\ f^2(x), \text { and } f^3(x)$ and see if a pattern emerges.  

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