Find the inverse function of the function f, of, x, equals, 4, cube root of, x, end cube rootf(x)=4

3

x

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Answer
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, cubed, divided by, 64, end fractionf
−1
(x)=
64
x
3



f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 64, x, cubedf
−1
(x)=64x
3

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, cubed, divided by, 4, end fractionf
−1
(x)=
4
x
3



f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 4, x, cubedf
−1
(x)=4x
3

1 answer

To find the inverse function of \( f(x) = 4 \sqrt[3]{x} \), we will follow these steps:

  1. Replace \( f(x) \) with \( y \): \[ y = 4 \sqrt[3]{x} \]

  2. Solve for \( x \) in terms of \( y \)**: \[ \frac{y}{4} = \sqrt[3]{x} \]

  3. Cube both sides to eliminate the cube root: \[ \left( \frac{y}{4} \right)^3 = x \]

  4. Simplify the expression: \[ x = \frac{y^3}{4^3} = \frac{y^3}{64} \]

  5. Replace \( y \) with \( x \) to denote the inverse function: \[ f^{-1}(x) = \frac{x^3}{64} \]

Thus, the inverse function is: \[ f^{-1}(x) = \frac{x^3}{64} \]

Answer

The correct choice is: \[ f^{-1}(x) = \frac{x^3}{64} \]