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

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

Step 2: Swap \(x\) and \(y\)
\[x = \sqrt[3]{y-2} + 8\]

Step 3: Solve for \(y\)
Subtracting 8 from both sides, we get:
\[x - 8 = \sqrt[3]{y-2}\]

Cubing both sides:
\[(x - 8)^3 = y - 2\]

Adding 2 to both sides:
\[(x - 8)^3 + 2 = y\]

Step 4: Replace \(y\) with \(f^{-1}(x)\)
\[\begin{aligned} f^{-1}(x) &= (x - 8)^3 + 2 \end{aligned}\]

Therefore, \(f^{-1}(x) = (x - 8)^3 + 2\) for the given function \(f(x) = \sqrt[3]{x-2} + 8\).