To find the inverse of the function [tex]\( f(x) = \sqrt[3]{x + 12} \)[/tex], we need to follow these steps:
1. Define the function: [tex]\( y = \sqrt[3]{x + 12} \)[/tex].
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: Inverse functions are essentially a reflection over the line [tex]\( y = x \)[/tex]. So, start by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \sqrt[3]{y + 12}.
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]: Solve the equation from step 2 for [tex]\( y \)[/tex]:
[tex]\[
x = \sqrt[3]{y + 12}.
\][/tex]
To isolate [tex]\( y \)[/tex], first cube both sides to get rid of the cube root:
[tex]\[
x^3 = y + 12.
\][/tex]
Then, solve for [tex]\( y \)[/tex]:
[tex]\[
y = x^3 - 12.
\][/tex]
4. Write the inverse function: The inverse function, denoted as [tex]\( f^{-1}(x) \)[/tex], is:
[tex]\[
f^{-1}(x) = x^3 - 12.
\][/tex]
Next, we compare this result to the given choices:
- A. [tex]\( f^{-1}(x) = 12 - x^3 \)[/tex]
- B. [tex]\( f^{-1}(x) = x - 12 \)[/tex]
- C. [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex]
- D. [tex]\( f^{-1}(x) = x + 12 \)[/tex]
The correct choice is clearly C: [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex].
