If I have two zero mean random variables say $x$ and $y$, and there is a function $f$ such that $$\mathrm{var}(x+f(x)) \approx \mathrm{var}(x)$$ and $$\mathrm{var}(y+f(y)) \approx \mathrm{var}(y)\; .$$
If we constrain $f$ such that $$E{f(x)}=E{f(y)}=0$$ and $x > f(x)$, for all $x$, Can I say that
$$E[(x+f(x))\cdot(y+f(y))] \approx E(xy) \; ?$$