The given problem asks to find the inverse of the function $f(x) = \frac{1}{2}x + 8$.

To find the inverse of a function, we follow these steps:

Steps to find the inverse of $f(x)$:

1. Start with the function $f(x) = \frac{1}{2}x + 8$.
2. Replace $f(x)$ with $y$, so:
   $y = \frac{1}{2}x + 8$
3. Swap $x$ and $y$ to find the inverse:
   $x = \frac{1}{2}y + 8$
4. Solve for $y$: $x - 8 = \frac{1}{2}y$ $y = 2(x - 8)$ $y = 2x - 16$
5. Thus, the inverse function is $f^{-1}(x) = 2x - 16$.

The correct answer is:

$\boxed{E. \ f^{-1}(x) = 2x - 16}$

Would you like more details or have any questions?
Here are 5 related questions:

1. How can you verify if two functions are inverses of each other?
2. What are the conditions for a function to have an inverse?
3. How do you find the inverse of a quadratic function?
4. Can a function have more than one inverse?
5. How does the graphical interpretation of inverse functions work?

Tip: To check if two functions are inverses, their compositions should give the identity function $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.