An rr function (i.e. *rational rational function*) is a quotient

$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$

such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$

**QUESTION** Do there exist rr functions $\ \phi\ \psi\ $
such that set

$$ \{(\phi(x),\ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq
        \ (\Bbb Q\cup\{\infty\})^2 $$

is dense in a non-empty open subset of
$\ \Bbb Q^2\,?$

I'd guess -- **NO**.