What functions do we need besides polynomials to describe any real as the root of some equation?

Question

If $f(x)$ is some polynomial with integer coefficient of degree $>0$, then any solution to $f(x) = 0$ is an algebraic number. If $g(x)$ and $h(x)$ are also polynomials with integer coefficients, is it possible for the solution of $f(x) +g(\sin x) = 0$ to be any real number? What about $f(x) + g(\sin x) + h(\log x) = 0$?

Answer 1

There is no way of constructing all the real numbers this way. Even if we have a countable set of special functions $\{f_n(x)\}_{n=1}^\infty$ and only allow arguments from any countable set (integers / rationals / algebraic numbers) then the set of numbers we can construct this way will be countable. The set of real numbers is uncountable so there will be real numbers we cannot get as $f_n(x)$ for $n,x\in\mathbb{N}$.