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The image you've uploaded appears to be focused on finding and verifying the inverse of a function \( f(x) \). I'll guide you step-by-step through solving and filling in the missing information.

### 1. **Interchanging \( x \) and \( y \):**
   The equation we start with is given as:
   \[
   y = \frac{x + 2}{2}
   \]
   Here, the goal is to find the inverse of \( f(x) \). To do this, we'll interchange \( x \) and \( y \), and then solve for \( y \).

### 2. **Replace \( y \) with \( f^{-1}(x) \):**
   After interchanging \( x \) and \( y \), we get:
   \[
   x = \frac{y + 2}{2}
   \]
   Now, solve for \( y \) (which will give us \( f^{-1}(x) \)):
   \[
   2x = y + 2 \quad \text{(Multiplying both sides by 2)}
   \]
   \[
   y = 2x - 2
   \]
   Therefore, the inverse function is:
   \[
   f^{-1}(x) = 2x - 2
   \]

### 3. **Verify that \( f^{-1}(f(x)) = x \) and \( f(f^{-1}(x)) = x \):**
   Let's first check \( f^{-1}(f(x)) \) and then \( f(f^{-1}(x)) \).

#### a) Verifying \( f^{-1}(f(x)) = x \):
   We know that:
   \[
   f(x) = \frac{x + 2}{2}
   \]
   So, apply \( f^{-1}(x) \) to \( f(x) \):
   \[
   f^{-1}(f(x)) = f^{-1}\left( \frac{x + 2}{2} \right)
   \]
   Substitute the formula for \( f^{-1}(x) \):
   \[
   f^{-1}\left( \frac{x + 2}{2} \right) = 2 \cdot \frac{x + 2}{2} - 2
   \]
   Simplify:
   \[
   2 \cdot \frac{x + 2}{2} = x + 2
   \]
   \[
   (x + 2) - 2 = x
   \]
   Thus, \( f^{-1}(f(x)) = x \).

#### b) Verifying \( f(f^{-1}(x)) = x \):
   We know that \( f^{-1}(x) = 2x - 2 \), so we substitute this into \( f(x) \):
   \[
   f(f^{-1}(x)) = f(2x - 2)
   \]
   Apply the formula for \( f(x) \):
   \[
   f(2x - 2) = \frac{(2x - 2) + 2}{2}
   \]
   Simplify:
   \[
   \frac{(2x - 2) + 2}{2} = \frac{2x}{2} = x
   \]
   Thus, \( f(f^{-1}(x)) = x \).

Both verifications are successful, confirming that \( f^{-1}(x) = 2x - 2 \) is correct.

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Would you like more details on any specific step, or have any questions?

### 5 Related Questions:
1. How do you find the inverse of more complicated functions?
2. What are the conditions for a function to have an inverse?
3. Can a function have more than one inverse?
4. How do you graph a function and its inverse on the same coordinate plane?
5. What are some real-world applications of inverse functions?

### Tip:
Always ensure the function you are working with is one-to-one (i.e., passes the horizontal line test) before attempting to find an inverse.

Interchange x and y. Replace y with f^{-1}(x). Verify that f^{-1}(f(x)) = x and f(f^{-1}(x)) = x. The function f(x) is given as y = (x + 2)/2.

Learn how to find the inverse of the function f(x) = (x + 2)/2 and verify that f^{-1}(f(x)) = x and f(f^{-1}(x)) = x. This guide covers the concepts of inverse functions, function composition, and algebraic manipulation. Suitable for high school students in Grades 10-12.

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