To find the inverse of the function $f(x) = \frac{6x}{x - 5}$, follow these steps:

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

We start by rewriting the equation as: $y = \frac{6x}{x - 5}$

Step 2: Swap $y$ and $x$

To find the inverse function, swap $y$ and $x$: $x = \frac{6y}{y - 5}$

Step 3: Solve for $y$

Now, solve this equation for $y$. Start by multiplying both sides of the equation by $y - 5$ to eliminate the denominator: $x(y - 5) = 6y$ Expand the left-hand side: $xy - 5x = 6y$ Move all terms involving $y$ to one side: $xy - 6y = 5x$ Factor out $y$ on the left-hand side: $y(x - 6) = 5x$ Solve for $y$: $y = \frac{5x}{x - 6}$

Step 4: Write the inverse function

The inverse function is: $f^{-1}(x) = \frac{5x}{x - 6}$

Final Answer:

$f^{-1}(x) = \frac{5x}{x - 6}$

Would you like to go over any details or have any questions?

Here are some related questions:

1. How do you verify the inverse of a function?
2. What are the domain and range of the inverse function?
3. Can all functions have inverses? Why or why not?
4. How would you graph the function and its inverse?
5. What are the conditions for a function to be one-to-one?

Tip: To check if two functions are inverses, you can compose them: $f(f^{-1}(x)) = f^{-1}(f(x)) = x$.