Related to $f$ equals <fill in blanks>
Is there a way to write a "function" (without lambda calculus) without a domain? For instance, when differentiating, one often just considers $x^3$ as a "universal function"; no domain is defined. And if we differentiate this, we get $3x^2$, no matter the domain:
$\begin{align}f_{\mathbb{R}}': &\quad\mathbb{R}\to \mathbb{R}\\&\quad x\mapsto 3x^2\end{align}$
$\begin{align}f_{\mathbb{N}}': &\quad\mathbb{N}\to \mathbb{N}\\&\quad x\mapsto 3x^2\end{align}$
$\begin{align}f_{\mathbb{C}}': &\quad\mathbb{C}\to \mathbb{C}\\&\quad x\mapsto 3x^2\end{align}$
However, treating $x^3$ as a "function" can be confusing, as it's not obvious too see the difference between itself (a "function") and the expression $x^3$. It seems obvious that there is some domain $\mathbb{U}$, which I believe would be sufficient to define such a "universal function".
$\begin{align}f_{\mathbb{U}}': &\quad\mathbb{U}\to \mathbb{U}\\&\quad x\mapsto 3x^2\end{align}$
However, this notation seems awkward, and is not suited for simple differentiation. And given that the domain isn't defined, it seems unnecessary to write it. So how would one write such a "function" without a specified domain?