David,
f(x) means that the function is in terms of x, but you're saying that the function f(x) is:
f(x) = (y+2)/(y^{2} - 2y)
which is in terms of y. So it should be either f(y) or f(x) = (x+2)/(x^{2} - 2x).
Domain, by definition, is a set of values in which the function is "defined".
First, factor the denominator x^{2} - 2x, which is x(x-2). And we know that denominators cannot be zero (or else the function is "undefined"). So this means that the value(s) of x that make(s) the function undefined is when...
x(x-2) = 0
This is true when x = 2 and when x = 0.
So the domain of the function is all values of x except when x = {0,2}. To write this is notation form D = (-infinity∪0∪2∪infinity).
I hope that helps! :)