The domain of a function is the set of possible values for the independent variable(s). In this case we are looking for the set of possible values for x. Generally it is easier to find what values of x are not possible, then give the answer as all real numbers not including the impossible values. So are there any values of x such that f(x) = 4 - x does not exist?
The range of a function is a little more difficult. The range of a function is the set of values that the function returns. (ie. what values can f(x) take?) My best recommendation is to graph the function (which is a downward opening parabola) and see what values of the function are allowed. So is there a lower limit for what f(x) = 2 - x^2 can be? Is there an upper limit?
For vertical asymptotes we are looking for values of x such that the denominator is 0.
For horizontal asymptotes we are looking for what the behavior of the function is for x tending toward positive and negative infinity. If the function approximates a constant for these limits, then it has a horizontal asymptote.
For symmetry about the x axis we are looking to see if for every point (x, f(x)) on the graph we also have the point (x, -f(x)) on the graph.
For symmetry about the line x = 3, the simplest thing to do is first consider y = f(x - 3). This means translate the function three units to the left. Then the question becomes is f(x - 3) symmetric about the line x = 0, aka the y axis? If a function g(x) is symmetric about the y axis then g(-x) = g(x). So the question, in its final form, becomes: Is f(-(x - 3)) = f(x - 3)?
Try your question with these and post what you come up with if you are still having problems.