# Is it possible to find the inverse of the sum of two functions, knowing the inverse of each?

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

• You have 13 now. ${}{}$ – user99914 Oct 8 '15 at 14:39
• What exactly are you trying to do? – rogerl Oct 8 '15 at 14:40
• Solution to what? – David C. Ullrich Oct 8 '15 at 14:44
• 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. – Paul Oct 8 '15 at 14:55
• 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?" – David C. Ullrich Oct 8 '15 at 15:55

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?