We are given the function $f(x) = \frac{5x - 7}{5x - 9}$, and we are tasked with finding the inverse of this function, $f^{-1}(x)$.

Steps to find the inverse:

1. Replace $f(x)$ with $y$: $y = \frac{5x - 7}{5x - 9}$

2. Swap $x$ and $y$: $x = \frac{5y - 7}{5y - 9}$

3. Solve for $y$: Multiply both sides by $5y - 9$ to eliminate the denominator: $x(5y - 9) = 5y - 7$ Expand both sides: $5xy - 9x = 5y - 7$ Get all terms involving $y$ on one side and other terms on the opposite side: $5xy - 5y = 9x - 7$ Factor out $y$ on the left-hand side: $y(5x - 5) = 9x - 7$ Solve for $y$: $y = \frac{9x - 7}{5x - 5}$

Thus, the inverse function is: $f^{-1}(x) = \frac{9x - 7}{5x - 5}$

Would you like a more detailed explanation of any steps?

Here are 5 related questions to explore:

1. How do you determine if a function has an inverse?
2. What is the graphical interpretation of finding the inverse of a function?
3. Are there any restrictions on the domain or range of the given function $f(x)$?
4. How would you verify that $f(f^{-1}(x)) = x$?
5. Can you find the inverse of a quadratic function using a similar process?

Tip: When solving for the inverse of a function, always make sure the original function is one-to-one (i.e., passes the horizontal line test).