Closed form for the inverse of $f(x) = x \log x$ over $[1, \infty]$? Consider
$$f(x) = x \log x$$
over $[1, \infty]$. The derivative is $1 + \log x$, which is positive, so the function is strictly increasing, hence an invertible map from $[1, \infty)$ to $[0, \infty)$ exists. But I'm having a hard time finding a way to express $f^{-1}$ (maybe there is no closed form?)      
 A: 
Maybe there is no closed form?

Yes. At least not in terms of elementary functions. However, you can express it in terms of certain special functions, such as Lambert's W function, in a way similar to that described in ‘Example $2$’.

An invertible map from $[1,\infty)$  to $[0,\infty)$  exists.

Just because something exists, doesn't mean that it can also be expressed in terms of other things. Fundamental colors also exist, but they cannot be expressed as a combination of other colors.

Can you give me some heuristics behind the intuition that this doesn't have a closed form ?

As far as “heuristics” and “intuition” are concerned, try comparing this situation with that of anti-derivatives; i.e., just because a function can be expressed as a combination of elementary functions does not mean that its primitive has the same property. Take, for instance, $e^{-x^2}$, whose integral is the non-elementary error function. But in order to actually be able to prove these facts, knowledge of abstract algebra is required.
