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July 13th, 2017, 12:47 PM   #1
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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.
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July 13th, 2017, 12:59 PM   #2
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Originally Posted by Mifarni14 View Post
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."
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July 16th, 2017, 12:16 PM   #3
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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.
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July 16th, 2017, 12:33 PM   #4
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Originally Posted by Mifarni14 View Post
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.
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August 18th, 2017, 12:03 PM   #5
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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.
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August 18th, 2017, 04:01 PM   #6
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The exercise is to calculate the image of f(x)
What did you find that image to be?
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August 19th, 2017, 04:45 AM   #7
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Hello,

I found it to be: $\displaystyle \frac{x}{\sqrt{1+2cx^2}}$
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August 19th, 2017, 08:01 AM   #8
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That is not what "image of a function" normally means.
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August 20th, 2017, 09:11 AM   #9
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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)).
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August 20th, 2017, 10:34 AM   #10
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Quote:
Originally Posted by Mifarni14 View Post
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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