To find the inverse of the function \( f(x) = (2x - 5)^3 \), we follow these steps:
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Replace \( f(x) \) with \( y \): \[ y = (2x - 5)^3 \]
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Switch \( x \) and \( y \) to find the inverse: \[ x = (2y - 5)^3 \]
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Solve for \( y \):
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Take the cube root of both sides: \[ \sqrt[3]{x} = 2y - 5 \]
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Add 5 to both sides: \[ \sqrt[3]{x} + 5 = 2y \]
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Divide by 2: \[ y = \frac{\sqrt[3]{x} + 5}{2} \]
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Write the inverse function: \[ f^{-1}(x) = \frac{\sqrt[3]{x} + 5}{2} \]
So, the inverse of the cube of \( 2x - 5 \) is: \[ f^{-1}(x) = \frac{\sqrt[3]{x} + 5}{2} \]