The domain of $\displaystyle f^{-1}$ is the range of f. (Going from f to $\displaystyle f^{-1}$ swaps "domain" and "range".) f(0)= 6 and as x goes to infinity, f(x) goes to negative infinity. So the domain of $\displaystyle f^{-1}$ is all number less than or equal to 6.
(The inverse function to $\displaystyle -x^2+ 6]$ is $\displaystyle \sqrt{-x+ 6}$, not $\displaystyle \sqrt{-x+ 3}$. Was that a typo?)