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Yesterday our analysis professor told us you cannot solve $$ y = e^x+2/(1+x^2) $$ for x, but you have the option to approximate this numerically. He did not prove that, he just noted it.

I can't believe that's true and am very unsatisfied with that. How do I solve this for x? If I don't, why is that?

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I'd like to see how an inverse function is constructed. –  Max Ried Nov 29 '12 at 8:01
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Proving that an equation cannot be solved using elementary functions is very difficult, and seldom done. There is moderately good theory (differential algebra) that deals with integration in elementary terms, and this can sometimes be adapted to inverse functions. From experience, one can guess that there is no elementary function $g$ such that $g(y)=x$. I doubt that this has ever been proved. –  André Nicolas Nov 29 '12 at 8:06

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What do you mean, you cant solve for $x$? given a fixed value of $y$, there will be some value that satisfies that equation. What he probably meant is you can't give the solution in explict/closed form. Which happens with many trancendetal equations, simply because we havn't defined an opperator/function to give the value of x in closed form. I don't know much on the topic but I think theres a series inversion formula known as "the lagrange inversion formula" that will give you the inverse, but I don't know how to use it. If you simply want to solve it, for a fixed value of $y$, you could try using newtons method or some other technique. In terms of proving it is unsolveable I wouldn't know how, nor would I think it would be very easy, as it is often very hard to prove general statements like that, but I know there are some fields, I think "Galois theory?", is one, that are devoted to primarly showing that certain equations can't be solved for, in terms of certain opperations, though, im quite sure most require a background in abstract algebra to understand.

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