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: \[ f^{-1}(x) = (x - 8)^3 + 2 \]