To find the inverse of the function \(f(x) = \sqrt[3]{x-14}\), follow these steps:
1. Express \(f(x)\) in terms of \(y\):
We write the original function as:
[tex]\[
y = \sqrt[3]{x - 14}
\][/tex]
2. Swap \(x\) and \(y\):
To find the inverse function, swap the variables \(x\) and \(y\):
[tex]\[
x = \sqrt[3]{y - 14}
\][/tex]
3. Solve for \(y\):
To isolate \(y\), first cube both sides of the equation to remove the cube root:
[tex]\[
x^3 = y - 14
\][/tex]
4. Isolate \(y\):
Now add 14 to both sides of the equation to solve for \(y\):
[tex]\[
y = x^3 + 14
\][/tex]
Thus, the inverse function is:
[tex]\[
f^{-1}(x) = x^3 + 14
\][/tex]
Given the answer choices:
- A. \(f^{-1}(x) = (x-14)^3\)
- B. \(f^{-1}(x) = x^3 + 14\)
- C. \(f^{-1}(x) = (x+14)^3\)
- D. \(f^{-1}(x) = x^3 - 14\)
The correct inverse function is:
[tex]\[
f^{-1}(x) = x^3 + 14
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{2}
\][/tex]
