It looks like you're working through the steps to solve for the inverse of the function, but there's a bit of confusion in how the terms are written and manipulated. Let's clarify the steps.

Original Function:

We start with the original function: $f(x) = \frac{x}{2} + 2$

Step-by-Step Explanation:

1. Replace $f(x)$ with $y$: $y = \frac{x}{2} + 2$ This step is simply renaming $f(x)$ as $y$.

2. Swap $x$ and $y$: $x = \frac{y}{2} + 2$ Here, you switch the roles of $x$ and $y$. The reason for this is that you're finding the inverse function, which essentially swaps the input and output.

3. Isolate $y$: To solve for $y$, you need to get $y$ by itself on one side of the equation: $x - 2 = \frac{y}{2}$ This step involves subtracting 2 from both sides of the equation to move the constant term to the left side.

4. Solve for $y$: Now, you solve for $y$ by multiplying both sides by 2: $y = 2(x - 2)$ This step isolates $y$, giving you the formula for the inverse function.

5. Simplify: Finally, you distribute the 2 on the right side: $y = 2x - 4$ This is your inverse function.

Summary

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

Each step involves basic algebraic manipulations: swapping variables, isolating the variable of interest, and simplifying the expression. The equation you mentioned:

$x - 2 = \frac{y}{2} \quad \text{and} \quad y = 2(x - 2)$

are intermediate steps to solve for $y$ in terms of $x$ when finding the inverse.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. Why is it necessary to swap $x$ and $y$ when finding the inverse of a function?
2. What are the implications of a function not having an inverse?
3. How can you graphically interpret the inverse of a function?
4. How do the domains and ranges of a function and its inverse relate to each other?
5. How do you find the inverse of a function that is not linear?

Tip: When solving for the inverse function, carefully track your algebraic steps to avoid common mistakes, like incorrect sign changes or improper distribution.