Find f^-1 for the function f(x) = (^3sqrtx-2)+8

Answer 1

To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = \sqrt[3]{x - 2} + 8 \), we follow these steps:

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

2. Solve for \( x \) in terms of \( y \): \[ y - 8 = \sqrt[3]{x - 2} \]

   Now, we will cube both sides to eliminate the cube root: \[ (y - 8)^3 = x - 2 \]

   Next, we isolate \( x \): \[ x = (y - 8)^3 + 2 \]

3. Replace \( y \) with \( x \) to write the inverse function: \[ f^{-1}(x) = (x - 8)^3 + 2 \]

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