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This problem was a question on a math test I took, and I didn't know how to solve it. How would you solve this?

Let $$f(x)=e^{-xe^{-\sqrt{x}}+e^{\sqrt{x}}}+2e^{xe^{-\sqrt{x}}+e^{\sqrt{x}}}.$$ Find $$f(f(f(f(f(f(f(f(f(f(x)))))))))).$$

EDIT

For more convenience, the expression of $f(x)$ is given by \begin{align*} f(x) = \exp\bigl(-x\exp(-\sqrt{x}) + \exp(\sqrt{x})\bigr)+{} \\{}+ 2\exp(x\exp(-\sqrt{x}) + +\exp\bigl(\sqrt{x})\bigr). \end{align*}

Michael Hoppe
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    Did you work out what $f(f(x))$ was? If it doesn't collapse, nothing comes to my mind right now. Off hand, this looks like a question designed to test whether you were wise enough to skip to the next. – Lubin Sep 06 '19 at 21:25
  • $f(f(0))$ is around $10^8$ and $f(f(f(0)))$ is too big for Maple. – GEdgar Sep 06 '19 at 21:31
  • @Lubin Are questions with the last purpose you stated intentionally designed in standard exams? If so I'm very surprised. It's the first time I'm hearing about a math exam testing for psychological skill. – Allawonder Sep 06 '19 at 21:44
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    Please give us the references of this math test. –  Sep 06 '19 at 22:21
  • Well, @Allawonder, this is not a standard question, either. Maybe it’s not to test psychological skill, but plain common sense. – Lubin Sep 06 '19 at 22:28
  • @Lubin LOL, well. Mayhaps I don't have plain common sense. Because if I'd had extra time left I've tried everything I knew on that problem... – Allawonder Sep 06 '19 at 22:32
  • I wouldn't be surprised if this somehow simplified to a hyperbolic trig function. – Andrew Chin Sep 07 '19 at 01:52
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    I would be extremely surprised if this problem has a solution better then brutally computing the composition. – Sangchul Lee Sep 07 '19 at 02:03
  • @GEdgar My CAS approximates it to $e^{4.948612527·10^{5061}}$ – Sudix Sep 07 '19 at 05:54
  • Observation: let $a=\exp(\sqrt{x})$. Then $$f(x)=e^a\bigl(2\cosh(x/a)+e^{x/a}\bigr).$$ – Michael Hoppe Sep 07 '19 at 16:45

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