To find the inverse of the function [tex]\( f(x) = \sqrt[3]{x + 12} \)[/tex], we need to follow these steps:
1. Express the function as an equation:
[tex]\[
y = \sqrt[3]{x + 12}
\][/tex]
2. Swap the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] to find the inverse. This means wherever there is [tex]\(y\)[/tex], we replace it with [tex]\(x\)[/tex], and wherever there is [tex]\(x\)[/tex], we replace it with [tex]\(y\)[/tex]:
[tex]\[
x = \sqrt[3]{y + 12}
\][/tex]
3. Solve for [tex]\(y\)[/tex] to get the inverse function:
- To eliminate the cube root, we cube both sides of the equation:
[tex]\[
x^3 = y + 12
\][/tex]
- Now, we need to isolate [tex]\(y\)[/tex]. Subtract 12 from both sides:
[tex]\[
y = x^3 - 12
\][/tex]
4. Write the inverse function:
[tex]\[
f^{-1}(x) = x^3 - 12
\][/tex]
By these steps, we determined that the inverse function is [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex].
Thus, the correct answer is:
B. [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex]
