$y=f\left(x\right)={(5-{(x-8)}^5)}^{\frac{1}{3}}$
then $y^3=5-(x-8{)}^5\Rightarrow (x-8{)}^5=5-y^3$
$\Rightarrow x=8+(5-y^3{)}^{\frac{1}{5}}$
Let, $z=g\left(x\right)=8+(5-x^3{)}^{\frac{1}{5}}$
To check, $f\left(g\right(x\left)\right)=[5-{(x-8)}^5{]}^{\frac{1}{3}}$
$={(5-{\left[{(5-x^3)}^{\frac{1}{5}}\right]}^5)}^{\frac{1}{3}}=(5-5+x^3{)}^{\frac{1}{3}}=x$
similarly , we can show that $g\left(f\right(x\left)\right)=x$
Hence, $g\left(x\right)=8+(5-x^3{)}^{\frac{1}{5}}$ is the inverse of $f\left(x\right)$