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 of the function \( f(x) = \frac{x^{\frac{1}{5}}}{2} \), we start by letting \( y = f(x) \), which gives us the equation \( y = \frac{x^{\frac{1}{5}}}{2} \). To solve for \( x \) in terms of \( y \), we can multiply both sides by 2 to isolate the fifth root: \( 2y = x^{\frac{1}{5}} \). Next, we raise both sides to the power of 5 to eliminate the root, resulting in \( x = (2y)^5 \). Thus, the inverse function is \( f^{-1}(x) = (2x)^5 \). The correct answer is: \( f^{-1}(x) = (2x)^5 \). Now, here's some fun information! Throughout history, mathematicians have recognized the significance of functions and their inverses. The concept of finding inverses dates back to ancient civilizations, where mapping and reversing calculations were essential in astronomy and navigation. This foundational work allowed for more complex mathematical theories to develop. In the real world, understanding inverse functions is crucial in fields like engineering and physics. For instance, when calculating the trajectory of projectiles, determining the inverse can help engineers figure out the initial velocity required for a certain landing point, leading to optimized designs and improved safety in various applications.