To determine the inverse function of \( f(x) = -x^3 - 9 \), we need to follow these steps:
1. Start with the original function:
[tex]\[
y = -x^3 - 9
\][/tex]
2. Swap \( x \) and \( y \) to find the inverse relation:
[tex]\[
x = -y^3 - 9
\][/tex]
3. Solve this equation for \( y \):
- First, isolate the term containing \( y \). Add 9 to both sides of the equation:
[tex]\[
x + 9 = -y^3
\][/tex]
- Next, multiply both sides by -1 to make the \( y^3 \) term positive:
[tex]\[
-(x + 9) = y^3
\][/tex]
- Finally, take the cube root of both sides to solve for \( y \):
[tex]\[
y = \sqrt[3]{-(x + 9)}
\][/tex]
Which can also be written as:
[tex]\[
y = -\sqrt[3]{x + 9}
\][/tex]
Therefore, the inverse function of \( f(x) = -x^3 - 9 \) is:
[tex]\[
f^{-1}(x) = -\sqrt[3]{x + 9}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{A. \ f^{-1}(x)=\sqrt[3]{x+9}}
\][/tex]
