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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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