So i found a question which states:
Without finding the inverse, state the domain and range of $f^{-1}$
$f(x) = (x-1)/(x-4)$
where $ x≠4$
how can i find the domain and range of the function's inverse without finding its inverse in the first place?
$\endgroup$ 11 Answer
$\begingroup$The domain of the inverse is the range of the original function, and vice versa (assuming the inverse exists, of course), since the inverse function is the reflection of the function over the line $y=x$.
Because the range of the function is $\mathbb R- \{1\}$ and the domain is $\mathbb R-\{4\}$, the inverse has domain $\mathbb R- \{1\}$ and range $\mathbb R-\{4\}$.
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