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.
(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**.