If [tex]f^{-1}(x)[/tex] is the inverse of [tex]f(x)[/tex], then
[tex]f\left(f^{-1}(x)\right) = x[/tex]
Given [tex]f(x)=x^3+9[/tex], we have
[tex]f\left(f^{-1}(x)\right) = \left(f^{-1}(x)\right)^3 + 9 = x[/tex]
Solve for the inverse :
[tex]\left(f^{-1}(x)\right)^3 + 9 = x \\\\ \left(f^{-1}(x)\right)^3 = x-9 \\\\ \boxed{f^{-1}(x) = \sqrt[3]{x-9}}[/tex]
Your answer is incorrect because it's interpreted as "3 times the square root of (x - 9)", whereas you want to end up with "the cube root of (x - 9)".
Try inserting ∛(x - 9) or (x - 9)^(1/3).
