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In the image, you're working on matching functions with their inverses. To find the inverse of a function \( f(x) \), we typically swap \( x \) and \( y \) in the equation for \( f(x) \), and solve for \( y \), which becomes the inverse function \( f^{-1}(x) \).

Let's start by analyzing one example and follow this process:

1. **Consider \( f(x) = \frac{2x - 1}{x + 2} \)**:
   - First, rewrite this as \( y = \frac{2x - 1}{x + 2} \).
   - Swap \( x \) and \( y \): \( x = \frac{2y - 1}{y + 2} \).
   - Now solve for \( y \):
     - Multiply both sides by \( (y + 2) \): \( x(y + 2) = 2y - 1 \).
     - Distribute \( x \): \( xy + 2x = 2y - 1 \).
     - Get all terms involving \( y \) on one side: \( xy - 2y = -1 - 2x \).
     - Factor out \( y \): \( y(x - 2) = -1 - 2x \).
     - Solve for \( y \): \( y = \frac{-1 - 2x}{x - 2} \).

So, the inverse of \( f(x) = \frac{2x - 1}{x + 2} \) is \( f^{-1}(x) = \frac{-1 - 2x}{x - 2} \).

Now, using this approach, you can match the rest of the functions with their inverses by following similar steps.

Would you like a detailed solution for each pair of functions? Feel free to ask!

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Here are five questions to help reinforce this topic:

1. What are the key steps for finding the inverse of a rational function?
2. How can you verify that two functions are inverses of each other?
3. Why do we switch \( x \) and \( y \) when finding the inverse of a function?
4. What are common mistakes to avoid when finding inverses of more complex functions?
5. How do you determine the domain and range of an inverse function?

**Tip:** When solving for the inverse, make sure to carefully handle any terms that involve both \( x \) and \( y \) on the same side of the equation!

Match the given functions with their inverses from the list.

Learn how to match rational functions with their inverses using algebraic techniques. This problem explores key concepts such as inverse functions, rational expressions, and their corresponding inverses. Suitable for high school students (Grades 9-12).

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