To find the inverse of the function $f(x) = \frac{-4x}{2x - 3}$, follow these steps:

Step 1: Replace $f(x)$ with $y$

Let: $y = \frac{-4x}{2x - 3}$

Step 2: Swap $x$ and $y$

To find the inverse, we swap $x$ and $y$: $x = \frac{-4y}{2y - 3}$

Step 3: Solve for $y$

Now, we solve for $y$ in terms of $x$.

Multiply both sides of the equation by $(2y - 3)$ to eliminate the denominator: $x(2y - 3) = -4y$

Distribute $x$ on the left-hand side: $2xy - 3x = -4y$

Move all terms involving $y$ to one side of the equation: $2xy + 4y = 3x$

Factor out $y$ on the left-hand side: $y(2x + 4) = 3x$

Now, divide both sides by $(2x + 4)$ to isolate $y$: $y = \frac{3x}{2x + 4}$

Step 4: Simplify the expression

We can simplify the denominator: $y = \frac{3x}{2(x + 2)}$

Thus, the inverse function is: $f^{-1}(x) = \frac{3x}{2(x + 2)}$

Final Answer:

The inverse of the function $f(x) = \frac{-4x}{2x - 3}$ is: $f^{-1}(x) = \frac{3x}{2(x + 2)}$

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Would you like more details or any further clarifications?

Here are some related questions to help expand on this concept:

1. How do you check if two functions are inverses of each other?
2. What is the domain of the inverse function in this case?
3. How would the inverse function change if the original function was modified?
4. Can every rational function have an inverse? Why or why not?
5. How do you find the derivative of an inverse function?

Tip: Always check that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ to confirm if the functions are truly inverses.