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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?

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Frank Vel
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  • I'm not sure you get anything sensible out of trying to do this. You have some idea of a domain in mind, since there are many sets where trying to define such a function is gibberish. – Malice Vidrine Jul 14 '16 at 15:44
  • @MaliceVidrine and there are countable many sets where such a function isn't gibberish. Too often "functions" are defined without a specific set in mind, so I think it would be nice to have a way to distinguish these from proper functions. – Frank Vel Jul 14 '16 at 15:46
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    These "definitions" are entirely informal; so it seems like what you're asking for is how to be more formal without resorting to the way these things are usually formalized. I don't think you're going to get better than $x\mapsto 3x^2$ or $\lambda x.3x^2$ without committing yourself to something more specific. – Malice Vidrine Jul 14 '16 at 16:54

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