The function \( f \) is one-to-one, Find its inverse, \( f(x)=(x-4)^{3} \) A. \( f^{-1}(x)=\sqrt[3]{x}+64 \) B. \( f^{-1}(x)=\sqrt{x}+4 \) C. \( f^{-1}(x)=\sqrt[3]{x}+4 \) D. \( f^{-1}(x)=\sqrt[3]{x}-4 \)

To find the inverse of the function \( f(x) = (x-4)^{3} \), you start by setting \( y = (x-4)^{3} \) and then solve for \( x \). 1. Rewrite the equation: \( y = (x-4)^{3} \) Now, take the cube root: \( \sqrt[3]{y} = x - 4 \) 2. Solve for \( x \): \( x = \sqrt[3]{y} + 4 \) Now replace \( y \) with \( x \) to express the inverse function: \( f^{-1}(x) = \sqrt[3]{x} + 4 \) Hence, the correct answer is: C. \( f^{-1}(x)=\sqrt[3]{x}+4 \) To verify, substitute \( f^{-1}(x) \) back into \( f(x) \) and check if you get the input \( x \) back. In this case, if you check: 1. Start with \( x = \sqrt[3]{y} + 4 \), plug it back into \( f\) to ensure \( f(f^{-1}(x)) = x \), confirming that your inverse is correct and well-defined! It’s a fun little trip to unravel functions and their inverses, like solving a puzzle where you bring things full circle!