*x*−2=2*y*​y=2(x−2)
*y*=2(*x*−2)

explain

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.

Explore the concept of inverse functions and learn how to find the inverse of a function step-by-step. This detailed explanation covers algebraic manipulations, including variable swapping and isolation techniques. Suitable for high school students (Grades 10-12).

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