Composition of function $f (x) = \sqrt{2x^2+17}-3$ a total of $10$ times

Question

So I have this question here. Obviously, I don't think I am expected to take a composition of a function 10 times. That would be crazy. My idea is that f(x) repeats 10 times and that $\sqrt{2}$ repeats 10 times too so I was thinking I could write $f(x)$ as:

$$f(x)=\sqrt{2^{(1/2)^{10}}(f(x))^{10}+17}-3.$$ I could then go from there or something. Anything to add? Am I completely off or am I doing something totally wrong?

Robert Z · Accepted Answer · 2016-09-11T07:26:33.203

3

Note that $f(2)=2$ and $f(4)=4$. Moreover $f$ is a continuous and strictly increasing function in $[2,4]$. These properties hold also for the composition $g$. Therefore $g((2,4))=(2,4)$ and by the Intermediate Value Theorem there is $c\in (2,4)$ such that $g(c)=3$. Note that since $g$ is strictly increasing the point $c$ is unique.

edited Sep 11 '16 at 07:26

answered Sep 11 '16 at 07:14

Robert Z

145,942

Rolle's theorem? – Future Math person Sep 11 '16 at 07:17
Yes. It holds for the composition as well. So in that case, doing it 10 times means it's still 2...
Therefore, the same holds for 4.... I can then use IVT to show there is a g(c) = 3?
– Future Math person Sep 11 '16 at 07:21
Ok. So use the Intermediate Value Theorem for $g$. – Robert Z Sep 11 '16 at 07:22
See my edit. Is that logic right? – Future Math person Sep 11 '16 at 07:23
@Subhashis Chakraborty Yes, it is right. – Robert Z Sep 11 '16 at 07:25

Composition of function $f (x) = \sqrt{2x^2+17}-3$ a total of $10$ times

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