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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) \( f^{-1}(x) = 2x - 6 \)**.

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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 \( f^{-1}(f(x)) = x \).

If the function f is defined by f(x) = (1/2)x + 3 for all x, which of the following defines the inverse of f?

(A) f⁻¹(x) = 2x - 6
(B) f⁻¹(x) = 2 / (x + 6)
(C) f⁻¹(x) = -(1/2)x - 3
(D) f⁻¹(x) = 2x + 1/3

Learn how to find the inverse of the linear function f(x) = (1/2)x + 3 with detailed steps and answer choices. This question focuses on inverse functions and key algebraic techniques, suitable for high school students (Grades 10-12).

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