Find the inverse function. \[ f(x)=x^{3}-7 \] \( f^{-1}(x)=[\sqrt[?]{x+[ }] \)

To find the inverse of the function \( f(x) = x^3 - 7 \), we start by replacing \( f(x) \) with \( y \): \[ y = x^3 - 7. \] Next, we solve for \( x \): \[ y + 7 = x^3 \] \[ x = \sqrt[3]{y + 7}. \] Now, we replace \( y \) with \( x \) to express the inverse function: \[ f^{-1}(x) = \sqrt[3]{x + 7}. \] To summarize, the inverse function is: \[ f^{-1}(x) = \sqrt[3]{x + 7}. \] Did you know that cubed functions can create some fascinating shapes in their graphs? For instance, the graph of \( f(x) = x^3 - 7 \) will be an S-curve that stretches between quadrants, reflecting all the input values naturally. It has a point where it intersects the x-axis at \( x = 2 \), meaning \( f(2) = 0 \). Inverse functions can be quite handy in real-life applications, such as in physics or economics! For example, if you know the volume of a cube and want to find the side length, you can use the inverse function derived from the volume formula \( V = s^3 \) (where \( s \) is the side length). Applying the inverse, where \( s = \sqrt[3]{V} \), lets you easily determine the dimensions you need!