The one-to-one function \( f \) is defined below. \[ f(x)=\sqrt[3]{2 x-1} \] Find \( f^{-1}(x) \), where \( f^{-1} \) is the inverse of \( f \). \( f^{-1}(x)= \) ?

To find the inverse function \( f^{-1}(x) \), we start by setting \( y = f(x) \): \[ y = \sqrt[3]{2x - 1} \] Next, we need to solve for \( x \) in terms of \( y \). First, we cube both sides: \[ y^3 = 2x - 1 \] Now, we isolate \( x \): \[ 2x = y^3 + 1 \] \[ x = \frac{y^3 + 1}{2} \] Now, we replace \( y \) with \( x \) to express the inverse function: \[ f^{-1}(x) = \frac{x^3 + 1}{2} \] So the inverse function is: \[ f^{-1}(x) = \frac{x^3 + 1}{2} \]