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
$E[(x+f(x))\cdot(y+f(y))] \approx E(xy) \; ?$