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 and state what we need to do:
We start with the function [tex]\( f(x) = \sqrt[3]{x + 12} \)[/tex]. To find its inverse, we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. So, we set [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = \sqrt[3]{x + 12}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
To isolate [tex]\( x \)[/tex], we first eliminate the cube root by cubing both sides of the equation:
[tex]\[
y^3 = (x + 12)
\][/tex]
3. Isolate [tex]\( x \)[/tex]:
Next, we solve for [tex]\( x \)[/tex] by subtracting 12 from both sides:
[tex]\[
x = y^3 - 12
\][/tex]
4. Rewrite the inverse function:
Since we started with [tex]\( y = f(x) \)[/tex] and isolated [tex]\( x \)[/tex], the equation we derived, [tex]\( x = y^3 - 12 \)[/tex], represents the inverse function. Therefore, we denote the inverse function as:
[tex]\[
f^{-1}(x) = x^3 - 12
\][/tex]
So, the correct inverse function is [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex].
Conclusion:
The correct answer is option D:
[tex]\[
f^{-1}(x) = x^3 - 12
\][/tex]
Therefore, the answer is [tex]\(\boxed{4}\)[/tex].
