To find the inverse of the function $f(x) = \frac{x}{2} - 4$, follow these steps:

Step 1: Replace $f(x)$ with $y$

Let $f(x) = \frac{x}{2} - 4$. First, rewrite the function as: $y = \frac{x}{2} - 4$

Step 2: Swap $x$ and $y$

To find the inverse, swap the variables $x$ and $y$. This will give you: $x = \frac{y}{2} - 4$

Step 3: Solve for $y$

Now, solve for $y$ to express the inverse function. Start by isolating the term involving $y$:

1. Add 4 to both sides: $x + 4 = \frac{y}{2}$

2. Multiply both sides by 2 to eliminate the denominator: $2(x + 4) = y$

Thus, the inverse function is: $f^{-1}(x) = 2(x + 4)$

Final Answer:

The inverse function is: $f^{-1}(x) = 2(x + 4)$

Would you like any further clarification or more detailed steps? Here are some related questions to explore:

1. How can we check that the inverse function is correct?
2. What does the graph of a function and its inverse look like?
3. How do we find the inverse of more complex functions involving squares or cubes?
4. What is the domain and range of the function and its inverse?
5. What would happen if we added another operation to the original function, like multiplying by a constant?

Tip: Always remember to swap the variables $x$ and $y$ when finding the inverse, and then solve for the new $y$ to get the inverse function.