The inverse [tex]f^{-1}(x)[/tex] is such that
[tex]f\left(f^{-1}(x)\right) = x[/tex]
We have
[tex]f\left(f^{-1}(x)\right) = -9\sqrt{f^{-1}(x) - 8} + 5 = x[/tex]
Solve for the inverse.
[tex]-9\sqrt{f^{-1}(x)-8} + 5 = x \\\\ -9\sqrt{f^{-1}(x)-8} = x-5 \\\\ \sqrt{f^{-1}(x)-8} = -\dfrac{x-5}9 \\\\ \left(\sqrt{f^{-1}(x)-8}\right)^2 = \left(-\dfrac{x-5}9\right)^2 \\\\ f^{-1}(x) - 8 = \dfrac{(x-5)^2}{81} \\\\ \boxed{f^{-1}(x) = 8 + \dfrac{(x-5)^2}{81}}[/tex]
