You actually only need to add one function to the set, but it's a very weird function. Define a function $f(R, x)$ such that $R$ is an encoding of any formula you like involving elementary functions, integrals, $x$ as a symbol, and $f$ itself (it is trivial to encode most parts of an elementary function in the integer part of $R$, and any real number constants in the function's formula can then be encoded as an index, and each of the real number constants' digits can then be alternated in fractional part of $R$; for instance, if you need to encode $a = a_1a_2a_3\ldots$ and $b = b_1b_2b_3\ldots$, the part of $R$ after the decimal point will look like $.a_1b_1a_2b_2a_3b_3\ldots$, and $a$ can be encoded as the index 1 and $b$ as the index 2), and $f(R, x)$ is defined as the result of evaluating the formula represented by $R$ on the input $x$.
This can also be done with two one-argument functions $f_1$ and $\textrm{spaceout}$, where $f_1$ behaves like $f$ except it takes its input with the digits alternating between the representations of $R$ and $X$, and the other, $\textrm{spaceout}(x)$, spaces the digits, so that for instance $13.12$ becomes $1030.102$; then, you can simply do the conversion as if you were using the two-argument form, and replace each $f(R, x)$ with $f_1(\textrm{spaceout}(R) + \textrm{spaceout}(x)/10)$
