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To find the inverse of the given function \( f(x) = \frac{-4x - 5}{4x - 3} \), follow these steps:

### Steps to find the inverse:

1. **Replace \( f(x) \) with \( y \)**:
   \[
   y = \frac{-4x - 5}{4x - 3}
   \]
   
2. **Swap \( x \) and \( y \)**:
   \[
   x = \frac{-4y - 5}{4y - 3}
   \]
   
3. **Solve for \( y \)**:
   Multiply both sides by \( 4y - 3 \) to eliminate the denominator:
   \[
   x(4y - 3) = -4y - 5
   \]
   
   Distribute \( x \):
   \[
   4xy - 3x = -4y - 5
   \]
   
4. **Group all terms involving \( y \) on one side**:
   \[
   4xy + 4y = 3x - 5
   \]
   
5. **Factor out \( y \)**:
   \[
   y(4x + 4) = 3x - 5
   \]
   
6. **Solve for \( y \)**:
   \[
   y = \frac{3x - 5}{4x + 4}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{3x - 5}{4x + 4}
\]

Let me know if you'd like more details on any step, and feel free to ask questions!

### Related Questions:
1. How do you verify if two functions are inverses of each other?
2. Can every function have an inverse? What are the conditions?
3. How does the behavior of a function change when you consider its inverse?
4. What does it mean graphically when a function and its inverse are plotted?
5. What role do one-to-one functions play in finding inverses?

### Tip:
When solving for the inverse, always ensure the function is one-to-one (passes the horizontal line test) to confirm that an inverse exists.

Find the inverse of f(x) = (-4x - 5) / (4x - 3).

Learn how to find the inverse of the function f(x) = (-4x - 5) / (4x - 3) with a detailed, step-by-step guide. This article covers essential concepts in functions and their inverses, tailored for college-level mathematics students.

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