To find the inverse function of [tex]\( f(x) = \sqrt[3]{x + 8} \)[/tex], let's follow the detailed steps:
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
Start by letting [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = \sqrt[3]{x + 8}
\][/tex]
2. Swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \sqrt[3]{y + 8}
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Now, solve the equation [tex]\( x = \sqrt[3]{y + 8} \)[/tex] for [tex]\( y \)[/tex]:
- Cube both sides to eliminate the cube root:
[tex]\[
x^3 = y + 8
\][/tex]
- Isolate [tex]\( y \)[/tex]:
[tex]\[
y = x^3 - 8
\][/tex]
4. Write the inverse function:
The inverse function [tex]\( f^{-1}(x) \)[/tex] is the expression we found for [tex]\( y \)[/tex]:
[tex]\[
f^{-1}(x) = x^3 - 8
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{A. \, f^{-1}(x) = x^3 - 8}
\][/tex]
