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To find the inverse of the function \( f(x) = -\frac{1}{2}(x + 3) \), follow these steps:

1. **Rewrite the function** in terms of \( y \):
   \[
   y = -\frac{1}{2}(x + 3)
   \]

2. **Swap** \( x \) and \( y \) to begin finding the inverse:
   \[
   x = -\frac{1}{2}(y + 3)
   \]

3. **Solve for** \( y \):
   \[
   x = -\frac{1}{2}y - \frac{3}{2}
   \]

   Multiply both sides by \(-2\) to eliminate the fraction:
   \[
   -2x = y + 3
   \]

   Subtract 3 from both sides:
   \[
   y = -2x - 3
   \]

So, the inverse function is:
\[
f^{-1}(x) = -2x - 3
\]

Would you like a detailed explanation or have any further questions?

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Here are 5 questions related to this topic:

1. What is the general method to find the inverse of a linear function?
2. How do you verify that two functions are inverses of each other?
3. What happens to the slope of the function when finding its inverse?
4. How does the domain and range of a function relate to its inverse?
5. What are the conditions for a function to have an inverse?

**Tip:** When finding the inverse, always check your final expression by plugging it back into the original function to see if you get the identity function \( f(f^{-1}(x)) = x \).

What is the inverse of the function f(x) = -1/2(x + 3)?

Learn how to find the inverse of the linear function f(x) = -1/2(x + 3). This problem covers key algebraic concepts such as inverse functions, linear equation manipulation, and the inverse function theorem. Suitable for students in Grades 9-11.

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