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?
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.