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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).

For the function f(x) = (5x - 7) / (5x - 9), find f^(-1)(x).

Learn how to find the inverse of the function f(x) = (5x - 7) / (5x - 9) with a detailed step-by-step explanation. This problem focuses on rational functions and inverse function theorem, suitable for students in Grades 10-12.

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