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For the function f, of, x, equals, 8, cube root of, x, end cube rootf(x)=8

3

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, find f, to the power minus 1 , left parenthesis, x, right parenthesisf
−1
(x)

3 answers

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

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

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

  3. Solve for \( y \):

    • First, divide both sides by 8: \[ \frac{x}{8} = \sqrt[3]{y} \]
    • Next, cube both sides to eliminate the cube root: \[ \left( \frac{x}{8} \right)^3 = y \]

  4. Express \( y \) in terms of \( x \): \[ y = \frac{x^3}{8^3} \] Since \( 8^3 = 512 \): \[ y = \frac{x^3}{512} \]

  5. Rewrite the equation to express the inverse function: \[ f^{-1}(x) = \frac{x^3}{512} \]

Thus, the inverse function is: \[ f^{-1}(x) = \frac{x^3}{512} \]

it says thats incorrect

Let’s go through the steps again to find the inverse function more carefully.

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

Step 1: Set the function equal to \( y \)

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

Step 2: Swap \( x \) and \( y \)

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

Step 3: Solve for \( y \)

  1. Divide both sides by 8: \[ \frac{x}{8} = \sqrt[3]{y} \]

  2. Cube both sides to eliminate the cube root: \[ \left( \frac{x}{8} \right)^3 = y \]

Step 4: Simplify the expression

\[ y = \frac{x^3}{8^3} \] Since \( 8^3 = 512 \), we have: \[ y = \frac{x^3}{512} \]

Step 5: Write the inverse function

So, the inverse function is: \[ f^{-1}(x) = \frac{x^3}{512} \]

Conclusion

The result \( f^{-1}(x) = \frac{x^3}{512} \) should indeed be correct based on the calculations performed. If there is an error being pointed out (such as in a specific context), please clarify or provide what the expected output is so I can address it accordingly.