Questions: Write an equation for the inverse of the given one-to-one function. f(x)=9-(x-8)^(1/3) f^(-1)(x)= -x^3+27x^2-243x+737 (Do not multiply. Do not combine like terms.)

Write an equation for the inverse of the given one-to-one function. f(x)=9-(x-8)^(1/3) f^(-1)(x)= -x^3+27x^2-243x+737 (Do not multiply. Do not combine like terms.)
Transcript text: Write an equation for the inverse of the given one-to-one function. \[ f(x)=9-\sqrt[3]{x-8} \] $f^{-1}(x)=$ $\square$ $-x^{3}+27 x^{2}-243 x+737$ (Do not multiply. Do not combine like terms.)
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Solution

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Solution Steps

To find the inverse of the given function \( f(x) = 9 - \sqrt[3]{x-8} \), we need to follow these steps:

  1. Replace \( f(x) \) with \( y \).
  2. Swap \( x \) and \( y \) to solve for \( y \) in terms of \( x \).
  3. Isolate \( y \) to express the inverse function \( f^{-1}(x) \).
Step 1: Define the Original Function

Given the function: \[ f(x) = 9 - \sqrt[3]{x - 8} \]

Step 2: Replace \( f(x) \) with \( y \)

\[ y = 9 - \sqrt[3]{x - 8} \]

Step 3: Swap \( x \) and \( y \)

\[ x = 9 - \sqrt[3]{y - 8} \]

Step 4: Isolate \( y \)

To isolate \( y \), we first move the constant term to the other side: \[ \sqrt[3]{y - 8} = 9 - x \]

Next, we cube both sides to remove the cube root: \[ y - 8 = (9 - x)^3 \]

Finally, we solve for \( y \): \[ y = (9 - x)^3 + 8 \]

Final Answer

\[ \boxed{f^{-1}(x) = (9 - x)^3 + 8} \]

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