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.

Question

(1 point)
Responses

lower f superscript negative 1 baseline left parenthesis x right parenthesis equals left parenthesis x minus 8 right parenthesis cubed plus 2
Image with alt text: lower f superscript negative 1 baseline left parenthesis x right parenthesis equals left parenthesis x minus 8 right parenthesis cubed plus 2

lower f superscript negative 1 baseline left parenthesis x right parenthesis equals left parenthesis x plus 8 right parenthesis superscript 3 baseline plus 2
Image with alt text: lower f superscript negative 1 baseline left parenthesis x right parenthesis equals left parenthesis x plus 8 right parenthesis superscript 3 baseline plus 2

lower f superscript negative 1 baseline left parenthesis x right parenthesis equals root index 3 Start Root x minus 8 End Root plus 2
Image with alt text: lower f superscript negative 1 baseline left parenthesis x right parenthesis equals root index 3 Start Root x minus 8 End Root plus 2

lower f superscript negative 1 baseline left parenthesis x right parenthesis equals left parenthesis x minus 8 right parenthesis superscript 3 baseline minus 2

Answer 1

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

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

2. Solve for \( x \) in terms of \( y \)**:

   • Subtract 8 from both sides: \[ y - 8 = \sqrt[3]{x - 2} \]
   • Cube both sides: \[ (y - 8)^3 = x - 2 \]
   • Add 2 to both sides: \[ x = (y - 8)^3 + 2 \]

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

So the correct expression for the inverse function is: \[ f^{-1}(x) = (x - 8)^3 + 2 \]

Thus, the response is: lower f superscript negative 1 baseline left parenthesis x right parenthesis equals left parenthesis x minus 8 right parenthesis cubed plus 2.