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

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

$y = \frac{4x}{x+5}$

Step 2: Swap $y$ and $x$:

$x = \frac{4y}{y+5}$ We want to solve for $y$ in terms of $x$.

Step 3: Multiply both sides by $(y+5)$ to eliminate the denominator:

$x(y + 5) = 4y$

Step 4: Distribute $x$ on the left side:

$xy + 5x = 4y$

Step 5: Move all terms involving $y$ to one side:

$xy - 4y = -5x$

Step 6: Factor out $y$ from the left side:

$y(x - 4) = -5x$

Step 7: Solve for $y$:

$y = \frac{-5x}{x - 4}$

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

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Would you like further details on any step or have any other questions?

Here are five related questions you might explore:

1. How do you verify if the inverse function is correct?
2. What happens if you try to find the inverse of a different rational function, like $f(x) = \frac{2x}{x-3}$?
3. How would you graph both $f(x)$ and $f^{-1}(x)$?
4. What is the domain and range of $f^{-1}(x)$?
5. Can all functions be inverted? Why or why not?

Tip: Always check the domain of the original function when finding its inverse, as it can affect the domain and range of the inverse function.