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

Step 1: Replace $f(x)$ with $y$

Let $y = \frac{x+2}{2x-1}$.

Step 2: Swap $x$ and $y$

To find the inverse, swap $x$ and $y$ and solve for $y$. So, we have: $x = \frac{y+2}{2y-1}$

Step 3: Solve for $y$

We now solve this equation for $y$.

1. Multiply both sides by $2y - 1$ to eliminate the denominator: $x(2y - 1) = y + 2$

2. Distribute $x$ on the left-hand side: $2xy - x = y + 2$

3. Move all terms involving $y$ to one side and constants to the other side: $2xy - y = x + 2$

4. Factor out $y$ on the left-hand side: $y(2x - 1) = x + 2$

5. Solve for $y$: $y = \frac{x + 2}{2x - 1}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x + 2}{2x - 1}$

Answer:

The inverse of $f(x) = \frac{x+2}{2x-1}$ is: $f^{-1}(x) = \frac{x+2}{2x-1}$

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Would you like further clarification on this or have any questions? Here are some related questions:

1. How can we verify that the inverse we found is correct?
2. How would you find the domain and range of the inverse function?
3. What is the significance of the composition $f(f^{-1}(x))$?
4. What are the steps involved in finding an inverse of a rational function?
5. How can you graph both a function and its inverse on the same coordinate plane?

Tip: When finding the inverse of a function, make sure to check the domain of the original function, as some values might be restricted when swapping $x$ and $y$.