Let $f(x)=\sqrt[\phantom{2}]{9-x^3},x0.$

$($a$)(3$ points$)$ What is the domain of $f^{-1}(x)?$

$($b$)(3$ points$)$ Find the value of $x$ such that $f(x)=1.$

$($c$)(5$ points$)$ Compute $(f^{-1}{)}^{\text{'}}(1).$