Select the correct answer. Find the inverse of the given function. \[ f(x)=\sqrt[3]{7 x-4} \]

To find the inverse of the function \( f(x) = \sqrt[3]{7x - 4} \), start by replacing \( f(x) \) with \( y \): \[ y = \sqrt[3]{7x - 4} \] Next, switch \( x \) and \( y \): \[ x = \sqrt[3]{7y - 4} \] Now, to eliminate the cube root, cube both sides: \[ x^3 = 7y - 4 \] Isolating \( y \) gives: \[ 7y = x^3 + 4 \] \[ y = \frac{x^3 + 4}{7} \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{x^3 + 4}{7} \] Fun fact: Inverse functions are like mirror images over the line \( y = x \! \). So if you take a point from \( f(x) \), reflect it over that line, and you'll land on the corresponding point of \( f^{-1}(x) \)! Understanding inverses can be super handy! For example, if you have a scenario where you know the output of a function and want to find the original input—like decoding a secret message or reversing a numerical transformation—being able to effectively find and use inverses opens up a lot of real-world possibilities!