To find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \sqrt[3]{2x - 1} \), follow these steps:
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Start by writing the function: \[ y = \sqrt[3]{2x - 1} \]
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Solve for \( x \) in terms of \( y \):
- First, cube both sides to eliminate the cube root: \[ y^3 = 2x - 1 \]
- Next, solve for \( 2x \): \[ 2x = y^3 + 1 \]
- Finally, solve for \( x \): \[ x = \frac{y^3 + 1}{2} \]
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Now, replace \( y \) with \( x \) to express the inverse: \[ f^{-1}(x) = \frac{x^3 + 1}{2} \]
Now, looking at the provided options:
- \( f^{-1}(x) = \frac{1}{2}(x^3 + 1) \) - This is equivalent to our derived equation.
- \( f^{-1}(x) = \frac{1}{2}(x + 1)^3 \) - This suggests a different expression.
- \( f^{-1}(x) = \left(\frac{x}{2} + 1\right)^3 \) - Also suggests a different expression.
- \( f^{-1}(x) = \left(\frac{x}{2}\right)^3 + 1 \) - Not consistent with our derived function.
The correct answer is: \[ f^{-1}(x) = \frac{1}{2}(x^3 + 1) \]