Since the domain of a polynomial function is $\mathbb{R}$, can we replace the input variable $x$ of any polynomial function by a one-to-one non-polynomial function (with range $\mathbb{R}$) and get another polynomial function?
For example, if $p(x)$ is a polynomial function, then can we always define $x$ to be a function such as $ $ $\sinh u$, so that $p(\sinh u)$ is a polynomial function?
Or in other words, is the following conjecture true:
For all polynomial functions $f(x)$ and for some function $g(x)$ with range and domain $R$, we can conclude that $f(g(x))$ is a polynomial function of $x$.
In general, the answer is "no": the composition of a polynomial function and a non-polynomial function of some argument $u$ is not a polynomial function in $u$. As the comments make clear, there are exceptions.
In your example, all one can say is: It's a polynomial in $\sinh u$, but not a polynomial in $u$. E.g. if $p(x) = x$ and $f(u) = \sinh u$, then $p\circ f(u) = \sinh u$, certainly not a polynomial in u (unless you consider the Taylor series as a polynomial of infinite degree: that was a fruitful avenue for Euler, but lesser mortals should probably avoid it). If you think that "polynomial in $\sinh u$" is an unnatural concoction, the notion of a "polynomial in some transcendental function" does come in handy in certain contexts, e.g. $\cos nx = p(\cos x)$ for some polynomial $p$ (but $\cos nx$ is certainly not a polynomial in $x$).
[Thanks to Ethan Bolker for pushing this answer and suggesting the simple example.]