[tex]f(x)=-x^3-9[/tex]
To find the inverse of f(x):
1. Substitute the f(x) by y:
[tex]y=-x^3-9[/tex]
2. Solve x:
Add 9 in both sides of the equation:
[tex]\begin{gathered} y+9=-x^3-9+9 \\ y+9=-x^3 \end{gathered}[/tex]
Multiply both sides of the equation by -1:
[tex]\begin{gathered} -1(y+9)=-1(-x^3) \\ -y-9=x^3 \end{gathered}[/tex]
Take cubic root in both sides of the equation:
[tex]\begin{gathered} \sqrt[3]{-y-9}=\sqrt[3]{x^3} \\ \\ x=\sqrt[3]{-y-9} \end{gathered}[/tex]
3.Change the y by x, and the x by f^-1 (x):
[tex]f^{-1}(x)=\sqrt[3]{-x-9}[/tex]
Then, the inverse of f(x) is B.
