unique solution of an algebraic system

Question

Suppose $a,b$ are real numbers satisfying $$ a = \frac{1}{2} (a+b) +\left(f(a)+f(b)\right)^{3/2}, $$ $$ b = \frac{1}{2} (a+b) -\left(f(a)+f(b)\right)^{3/2}. $$ Under what conditions on $f$ are $a$ and $b$ uniquely determined from the above.
Suppose for example that $f$ is monotone in a neighborhood of 0, do there exist unique $a$ and $b$?