46. Given the function \( f(x)=x^{3}-6 \), find \( f^{-1}(x) \). \( \begin{array}{ll}\text { a) } f^{-1}(x)=x^{1 / 3}+6 & \text { b) } f^{-1}(x)=\frac{x^{1 / 3}+6}{6} \\ \text { c) } f^{-1}(x)=(x+6)^{3} & \text { d) } f^{-1}(x)=(x+6)^{1 / 3}\end{array} \)

To find the inverse function \( f^{-1}(x) \) of \( f(x) = x^3 - 6 \), we start by setting \( y = f(x) \), which gives us the equation \( y = x^3 - 6 \). To solve for \( x \) in terms of \( y \), we can rearrange this to \( x^3 = y + 6 \). Then, taking the cube root of both sides, we have \( x = (y + 6)^{1/3} \). Thus, the inverse function is \( f^{-1}(x) = (x + 6)^{1/3} \), which corresponds to option d. In simpler terms, if you ever lose your way solving for an inverse, just remember: isolate your variable first! It’s like making a delicious cake—measure the ingredients (in this case, the variables) before mixing! Moreover, getting familiar with inverse functions can help in real-world applications such as calculating the original input from an output in various fields like engineering, economics, or even just for fun with your favorite games! You might find it interesting how versatile these mathematical concepts are!