f(x) = 4/(x - 9) - 12
The domain is the set of all possible x-values; all x-values are legal except for 9 (which makes the denominator 0).
Every y-alue is possible except for -12 (you can't get -12 because: the only way that you can get -12 is for 4/(x - 9) to be 0 and you can't find an x-value that works.)
Another way to look at this: substitute y for f(x)
---> y = 4/(x - 9) - 12
Add 12 to both sides:
---> y + 12 = 4/(x - 9)
Multiply both sides by x - 9:
---> (x - 9)(y + 12) = 4
Neither x - 9 nor y + 12 can be 0 (if they were, the answer would be 0, not 4)
So, x can't be 9 and y can't be -12. (Anything else is OK.)