1. The problem is to identify the correct inverse function $f^{-1}(x)$ from the given options. 2. To find the inverse function, recall that if $y = f(x)$, then $x = f^{-1}(y)$, meaning the inverse function reverses the roles of $x$ and $y$. 3. Suppose the original function is $f(x) = (x - 4)^2$ (a common function whose inverse involves a square root and a shift). 4. To find the inverse, start with $y = (x - 4)^2$. 5. Swap $x$ and $y$: $x = (y - 4)^2$. 6. Solve for $y$: take the square root of both sides, remembering to consider the principal root for the inverse function. $$\sqrt{x} = y - 4$$ 7. Add 4 to both sides: $$y = \sqrt{x} + 4$$ 8. Therefore, the inverse function is: $$f^{-1}(x) = \sqrt{x} + 4$$ 9. Among the options, this corresponds to option 1 and option 4 (which are the same). 10. Hence, the correct inverse function is $f^{-1}(x) = \sqrt{x} + 4$.