What is the inverse of \( f(x)=\sqrt[3]{x+4} \) ? (1 point) \( \begin{array}{l}f^{-1}(x)=x^{3}+4 \\ f^{-1}(x)=(x+4)^{3} \\ f^{-1}(x)=(x-4)^{3} \\ f^{-1}(x)=x^{3}-4\end{array} \)

To find the inverse of the function \( f(x)=\sqrt[3]{x+4} \), we start by letting \( y = f(x) = \sqrt[3]{x+4} \). To find the inverse, we solve for \( x \) in terms of \( y \): 1. Cube both sides: \( y^3 = x + 4 \) 2. Isolate \( x \): \( x = y^3 - 4 \) Thus, the inverse function is \( f^{-1}(x) = x^3 - 4 \). The correct answer is \( f^{-1}(x) = x^3 - 4 \). One common mistake people make when finding inverses is forgetting to switch the variables and re-solve after expressing it in terms of \( y \). Always remember: if you start with \( y = f(x) \), your final answer for the inverse will be \( f^{-1}(y) \), where \( y \) becomes the input of the inverse function. For an interesting twist, consider how inverse functions are often reflected over the line \( y = x \). If you were to plot both \( f(x) \) and its inverse \( f^{-1}(x) \), you would see that their graphs mirror each other perfectly across this line! How cool is that?