I am wondering if there exists an inverse function for $\lceil{e^{x}}\rceil$ over the natural numbers. I don't think it is a trivial task to derive an inverse function for a function containing a ceiling or a floor even if you know there is a bijection, and that an inverse function does exist. Am I wrong?
Thanks for any insight...
On
Yes, if we regard $x \mapsto \lceil e^x \rceil$ as a function with codomain its image, it is bijective, and this doesn't depend on whether we include $0$ in $\mathbb{N}$ or not. For $x \in [0, \infty)$, $\frac{d}{dx} e^x = e^x \geq 1$, and so $$e^{x + 1} = e^x + \int_x^{x + 1} e^x \, dt \geq e^x + \int_x^{x + 1} \,dt = e^x + 1.$$ In particular, $$\lceil e^{x + 1} \rceil > \lceil e^x \rceil,$$ so the function is injective. In particular it is bijective onto its image and so admits an inverse.
While that function is injective, it is not surjective, so it can't has an inverse in the common sense.
What it does have is a right inverse, that is, a function $g$ such that $f \circ g = id$, and you can simply define it as $g(x) = y$ such that $f(y) =x$, which is guaranteed existing a unique.
Finding a closed form for that is another thing, and in general it's not possible, in this case I don't know.