RU-SPB 2021 Problem 114

The following functions are written on the board, $$F(x) = x^2 + \frac{12}{x^2}, G(x) = \sin(\pi x^2), H(x) = 1.$$ If functions $f,g$ are currently on the board, we may write on the board the functions $$f(x) + g(x), f(x) - g(x), f(x)g(x), cf(x)$$ (the last for any real number $c$ ). Can a function $h(x)$ appear on the board such that $$|h(x) - x| < \frac{1}{3}$$ for all $x \in [1,10]$ ?