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 i want to find the domain of the inverse
There's essential two ways to find the domain of the inverse.

1. Solve for the inverse $$f^{-1}$$. Then find the vertical asymptote(there's one in the inverse function). Find the domain of $$f^{-1}$$
2. Find the range of the original function f. Find the horizontal asymptote in f. Find the range of the f= domain of $$f^{-1}$$

I do not get the same inverse function as you did. Show how you got that answer.

 $$y = -\frac{6}{x-3}$$
Even if you get the inverse function to be this, how did you get the range to be y<6? There's a horizontal asymptote at y=0