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!

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!