Question

Find f superscript negative 1 baseline for the function lower f left parenthesis x right parenthesis equals root index 3 Start Root x minus 2 End Root plus 8.

Answer 1

To find the inverse of the function \( f(x) = \sqrt[3]{x - 2} + 8 \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = \sqrt[3]{x - 2} + 8
\]

2. **Isolate \( \sqrt[3]{x - 2} \)**:
\[
y - 8 = \sqrt[3]{x - 2}
\]

3. **Cube both sides to eliminate the cube root**:
\[
(y - 8)^3 = x - 2
\]

4. **Solve for \( x \)**:
\[
x = (y - 8)^3 + 2
\]

5. **Finally, replace \( y \) with \( f^{-1}(x) \)**:
\[
f^{-1}(x) = (x - 8)^3 + 2
\]

Thus, the inverse function is:
\[
f^{-1}(x) = (x - 8)^3 + 2
\]