To find the inverse function of $f(x) = \frac{5x - 1}{x + 2}$, follow these steps:

1. Express $y = f(x)$:

$y = \frac{5x - 1}{x + 2}$

2. Swap $y$ and $x$:

$x = \frac{5y - 1}{y + 2}$

3. Solve for $y$:

Multiply both sides by $y + 2$ to eliminate the denominator: $x(y + 2) = 5y - 1$ Expand and rearrange to group terms involving $y$: $xy + 2x = 5y - 1$ Move all terms involving $y$ to one side: $xy - 5y = -2x - 1$ Factor out $y$: $y(x - 5) = -2x - 1$ Finally, solve for $y$: $y = \frac{-2x - 1}{x - 5}$

4. State the inverse function:

$f^{-1}(x) = \frac{-2x - 1}{x - 5}$

Would you like a deeper explanation, or do you have any questions?

Here are some related questions:

1. How can you verify if a function and its inverse are correct?
2. What is the domain of the inverse function $f^{-1}(x) = \frac{-2x - 1}{x - 5}$?
3. How does the horizontal line test relate to finding inverse functions?
4. What is the significance of the inverse function in real-world applications?
5. What happens if the original function is not one-to-one?

Tip: Always check if a function is one-to-one (injective) before attempting to find its inverse.