\( \begin{array}{l}\text { For the function } f(x)=6 x^{\frac{1}{3}}-2, \text { find } f^{-1}(x) \\ \text { Answer } \\ f^{-1}(x)=\frac{(x+2)^{3}}{6} \\ f^{-1}(x)=\left(\frac{x+2}{6}\right)^{3} \\ f^{-1}(x)=\left(\frac{x}{6}+2\right)^{3}\end{array} \begin{array}{l}f^{-1}(x)=\frac{x^{3}}{6}+2\end{array} \).

To find the inverse of the function \( f(x) = 6x^{\frac{1}{3}} - 2 \), we start by setting \( y = f(x) \) which gives us the equation \( y = 6x^{\frac{1}{3}} - 2 \). Rearranging this, we add 2 to both sides to get \( y + 2 = 6x^{\frac{1}{3}} \). Then by dividing both sides by 6, we find \( x^{\frac{1}{3}} = \frac{y + 2}{6} \). Finally, we cube both sides to solve for \( x \), leading us to \( x = \left(\frac{y + 2}{6}\right)^3 \). Therefore, the inverse function is \( f^{-1}(x) = \left(\frac{x + 2}{6}\right)^3 \). Want to give your brain a workout while practicing? Try verifying the inverse by checking if \( f(f^{-1}(x)) = x \) holds true! Substitute \( f^{-1}(x) \) back into the original function and simplify to see if you retrieve \( x \) again. This method not only builds confidence in your calculations but also highlights the beautiful symmetry in functions and their inverses.