https://file.aiquickdraw.com/mathbot/file/1736291976450pmkh9dvz.jpg

To find the inverse of the function in the given image, let’s break down the process step by step. The function provided is:

\[
f(x) = \frac{-3x - 4}{2 - x}
\]

We will determine the inverse \( f^{-1}(x) \) as follows:

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### Steps to Find the Inverse:

1. **Replace \( f(x) \) with \( y \):**
   \[
   y = \frac{-3x - 4}{2 - x}
   \]

2. **Swap \( x \) and \( y \):**
   This step is crucial because the inverse function reflects \( x \) and \( y \) in the equation:
   \[
   x = \frac{-3y - 4}{2 - y}
   \]

3. **Solve for \( y \):**
   Rearrange the equation to isolate \( y \):
   - Multiply both sides by \( (2 - y) \) to remove the denominator:
     \[
     x(2 - y) = -3y - 4
     \]
   - Expand:
     \[
     2x - xy = -3y - 4
     \]
   - Move all terms involving \( y \) to one side:
     \[
     2x + 4 = xy - 3y
     \]
   - Factor out \( y \) on the right-hand side:
     \[
     2x + 4 = y(x - 3)
     \]
   - Solve for \( y \):
     \[
     y = \frac{2x + 4}{x - 3}
     \]

---

### Final Answer:
The inverse function is:
\[
f^{-1}(x) = \frac{2x + 4}{x - 3}
\]

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Let me know if you'd like me to verify the result or if you have any additional questions! Below are some related problems for practice.

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### Related Questions:
1. How can you confirm that \( f(x) \) and \( f^{-1}(x) \) are truly inverses of each other?
2. What is the domain of \( f(x) = \frac{-3x - 4}{2 - x} \)?
3. What is the domain of \( f^{-1}(x) = \frac{2x + 4}{x - 3} \)?
4. If \( f(x) = \frac{-3x - 4}{2 - x} \), what is \( f(1) \)?
5. How does the graph of \( f(x) \) relate to the graph of its inverse?

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### Tip:
When solving for the inverse, always check for restrictions in the domain to avoid division by zero or other undefined operations.

Learn how to find the inverse of the rational function f(x) = (-3x - 4) / (2 - x) with this step-by-step guide. Master concepts such as rational equations, function inverses, and domain restrictions. Suitable for high school students and above.

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