My Math Forum Generalizing with a function

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 July 13th, 2017, 12:47 PM #1 Member   Joined: Sep 2014 From: Morocco Posts: 33 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
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 Originally Posted by Mifarni14 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.
I think they want you to plug the actual expression for $f$ into the iterated forms and see if you can find any interesting simplifications.

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: 33 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
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 Originally Posted by Mifarni14 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.
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: 33 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
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 Originally Posted by Mifarni14 The exercise is to calculate the image of f(x)
What did you find that image to be?

 August 19th, 2017, 04:45 AM #7 Member   Joined: Sep 2014 From: Morocco Posts: 33 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: 2,649 Thanks: 680 That is not what "image of a function" normally means.
 August 20th, 2017, 09:11 AM #9 Member   Joined: Sep 2014 From: Morocco Posts: 33 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
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Quote:
 Originally Posted by Mifarni14 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.
In English, f(g(x)) is called a composition of functions. I am guessing that what you are talking about might be called an iterated composition.

I am going to alter the notation a bit.

$f^0(x) = x.$

$f^n(x) = \dfrac{f^{(n-1)}(x)}{\sqrt{1 + c * \{f^{(n-1)}(x)\}^2}} \text { if } n \ge 1.$

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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