The problem provides a function $f(x) = \frac{1}{2}x + 3$ and asks for its inverse.

Steps to Find the Inverse of $f(x)$:

1. Rewrite $y = f(x)$:
   $y = \frac{1}{2}x + 3$

2. Swap $x$ and $y$:
   The inverse function switches the roles of $x$ and $y$.
   $x = \frac{1}{2}y + 3$

3. Solve for $y$:
   $x - 3 = \frac{1}{2}y$
   Multiply both sides by 2:
   $2(x - 3) = y$
   $y = 2x - 6$

4. Write the inverse function:
   $f^{-1}(x) = 2x - 6$

Correct Option:

The correct answer is (A) $f^{-1}(x) = 2x - 6$.

Do you need further details or have any questions?
Here are five related questions to explore:

1. How do you verify that two functions are inverses of each other?
2. What is the general process for finding an inverse function?
3. Why do we swap $x$ and $y$ when finding the inverse?
4. What are the conditions for a function to have an inverse?
5. How do inverse functions behave graphically?

Tip: To verify an inverse function, check if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.