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I'm probably trying the impossible, but do hope you can find a solution.

The problem arises since this function is not invertible $$ \sqrt{\frac{10-x}{4x}} + (2.5\pi-5\arctan{\sqrt{\frac{10-x}{x}}}) \tag{1}$$

but the two separate elements are invertible (I'll post the links and a picture when I have 10 rep) and give the inverse first, and second function $$5-\sqrt(25-4x^2)+ \frac{10}{(\tan^2(\pi/2-.2x) +1)}$$

When, for example, $f(x) = 5\rightarrow x= 6.42699$ and that value is the sum of respectively: 2.5 + 2.92699 (you can check: substituting those values, we get $y=5$ in both elements.

Is there any stratagem to find $f(x)$?

I'd appreciate it very much if you can tell me if I should stop trying (I am a student and I have been working over a month on this) or if you can help me find the solution, if there is one.

Edit:

I can't add a picture, so I uploaded it on a web-site: picture A the 2 curves in blue are the elements of the non-invertible function: the curve in red is the curve produced by the two elements of (1) when the axes have been swapped.

If, as it seems, there is no way to

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  • $\begingroup$ You have 13 now. ${}{}$ $\endgroup$ – user99914 Oct 8 '15 at 14:39
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    $\begingroup$ What exactly are you trying to do? $\endgroup$ – rogerl Oct 8 '15 at 14:40
  • $\begingroup$ Solution to what? $\endgroup$ – David C. Ullrich Oct 8 '15 at 14:44
  • $\begingroup$ I think he's trying to find the closed form of the inverse. In other words the formula for the red graph in the link he provided. $\endgroup$ – Paul Oct 8 '15 at 14:55
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    $\begingroup$ Yes I've seen the picture! The picture does not say what you're trying to do! I don't think you're intentionally being difficult, but I have a hard time seeing what you don't get about the question "what are you trying to do?" $\endgroup$ – David C. Ullrich Oct 8 '15 at 15:55
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Sums of invertible functions are in general not invertible.

For example, $f(x)=x$ and $g(x)=-x$ are invertible but $f(x)+g(x)=0$ clearly isn't.

But two functions $f(x)$ and $g(x)$ make for an invertible sum $f(x)+g(x)$ on open intervals where $f^\prime(x)\ne -g^\prime(x)$ following the inverse function theorem. Can you see where that fails to hold in your example?

Now, you seem to be interested in numerical values. Why don't you try different values of x in your first function and refine your search at each step? This should lead you to a suitable solution with precision chosen by you.

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  • $\begingroup$ You would like an equation for the inverse of your first function? This cannot be done since it is not invertible. You could produce a table with x and their corresponding y. Then switch their places and plot that function. $\endgroup$ – Jean-François Gagnon Oct 8 '15 at 16:27
  • $\begingroup$ You may want to have more points x where the derivative is large since the function is varying quickly there. $\endgroup$ – Jean-François Gagnon Oct 8 '15 at 16:27
  • $\begingroup$ It is possible that such a function does not exist. Try searching for "best function to approximate list of points". Maybe that could help. $\endgroup$ – Jean-François Gagnon Oct 8 '15 at 16:56

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