The domain of a function is the set of possible values you can input into it. In this case, the only value that cannot be entered into f(x) is 9, as that would result in f(9) - 4/(9-9) = 4/0. Thus the domain of f(x) is {x in RR|x != 9}
The range of f(x) is the set of all possible outputs of the function. That is, it is the set of all values which can be obtained by inputting something from the domain into f(x). In this case, the range consists of all real numbers besides 0, as for any nonzero real number y in RR, we can input (9y-4)/y into f and obtain
f((9y-4)/y) = 4/(9-(9y-4)/y) = (4y)/(9y - 9y + 4) = (4y)/4 = y
The fact that this works shows that f^(-1)(y) = (9y-4)/y is actually the inverse function of f(x). It turns out that the domain of the inverse function is the same as the range of the original function, meaning that the range of f(x) is the set of possible values you can input into f^(-1)(y) = (9y-4)/y. As the only value that cannot be entered into this is zero, we have the desired range as
{x in RR|x!=0}
