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The image displays a math problem asking to find the inverse of a function \( f(x) = \frac{5x + 4}{2x - 1} \). Let's solve this problem step by step.

### Steps to find the inverse of the function \( f(x) = \frac{5x + 4}{2x - 1} \):

1. **Set \( y = f(x) \):**
   \[
   y = \frac{5x + 4}{2x - 1}
   \]

2. **Swap \( x \) and \( y \):**
   \[
   x = \frac{5y + 4}{2y - 1}
   \]

3. **Solve for \( y \):**
   Multiply both sides by \( 2y - 1 \) to eliminate the denominator:
   \[
   x(2y - 1) = 5y + 4
   \]

   Distribute \( x \) on the left side:
   \[
   2xy - x = 5y + 4
   \]

4. **Collect all terms involving \( y \) on one side:**
   \[
   2xy - 5y = x + 4
   \]

5. **Factor out \( y \) on the left-hand side:**
   \[
   y(2x - 5) = x + 4
   \]

6. **Solve for \( y \):**
   \[
   y = \frac{x + 4}{2x - 5}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x + 4}{2x - 5}
\]

Would you like further clarification or additional details?

Here are some related questions you could explore:
1. How do you find the inverse of more complex rational functions?
2. What is the domain and range of the inverse function?
3. What is the significance of finding inverse functions in real-world applications?
4. How do vertical and horizontal asymptotes behave in inverse functions?
5. What happens when the original function is not one-to-one? How does it affect the inverse?

**Tip:** Always check the domain and range of both the function and its inverse to avoid undefined values.

Find the inverse of the following function: f(x) = (5x + 4) / (2x - 1)

Learn how to find the inverse of the rational function f(x) = (5x + 4) / (2x - 1) with a detailed step-by-step solution. This problem explores inverse functions, algebraic manipulation, and rational expressions, suitable for students in Grades 10-12.

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