"Least trivial" function preserving rationality Is there a "non-trivial" function $f(x,y)$ such that 

$$f(x,y) \in \mathbb{Q} \iff x,y\in \mathbb{Q}?$$

An example of a "trivial" function would be 

$$f(x,y) = \begin{cases} 0 &amp; x,y\in \mathbb{Q}\\ \pi &amp; \text{otherwise} \end{cases}$$ 
or any other $f$ which effectively uses a cases function.

The motivation is just my curiosity. Obviously, operations which preserve one direction of the $\iff$ are plentiful and well-studied. I was wondering how onerous the condition of the additional direction is on the choice of $f$. This question on mathoverflow seems related.
Perhaps a slightly less trivial example would be the function $f$ that interleaves the decimal digits of $x$ and $y$: that is, if
\begin{align}
x&amp;=0.x_1x_2x_3\dots \\
y&amp;=0.y_1y_2y_3\dots
\end{align}
are decimal expansions of $x$ and $y$, then
$$
f(x,y)=0.x_1y_1x_2y_2x_3y_3\dots
$$

You can make this definition unambiguous by deciding to always (or never) take finite decimal expansions when they are available.

Then $f(x,y)$ has a repeating decimal expansion if and only if both $x$ and $y$ do, and so it satisfies your condition.
I think the function f must be understood to be from $\mathbb{R}^2$ to $\mathbb{R}$ so I choose