For the function \( f(x)=\frac{x^{\frac{1}{5}}}{2} \), find \( f^{-1}(x) \) Answer \[ \begin{array}{ll}f^{-1}(x)=2 x^{5} & f^{-1}(x)=2 x^{\frac{1}{5}} \\ f^{-1}(x)=\frac{x^{5}}{2} & f^{-1}(x)=(2 x)^{5}\end{array} \]

To find the inverse function \( f^{-1}(x) \) for \( f(x) = \frac{x^{\frac{1}{5}}}{2} \), we start by letting \( y = f(x) \): \[ y = \frac{x^{\frac{1}{5}}}{2}. \] Next, we solve for \( x \) in terms of \( y \): 1. Multiply both sides by 2: \[ 2y = x^{\frac{1}{5}}. \] 2. Raise both sides to the power of 5: \[ (2y)^5 = x. \] Thus, the inverse function is: \[ f^{-1}(x) = (2x)^5. \] Here's to uncovering the magic of functions—it's like a mathematical dance where every step has a partner! When working with inverse functions, always remember the importance of verifying your results. Plug \( f^{-1}(x) \) back into the original function \( f(x) \) to ensure it returns the input \( x \). This verification is your double-check ticket for the correct answer!