The domain of h(x) is the set of all values of x that generate a sensible result in h. Most often, assume all real numbers as the domain and then we look for exceptions. Exceptions are usually revealed by such things as values of x that cause division by zero errors or square roots of negative numbers.
Remember that you shouldn't "simplify" fractions in the defined function before looking for exceptions. For example, if f(x) = x2/x then the domain does not include zero, because that would make the denominator zero. If you simplified first and got f(x) = x, then zero would be allowed. The trouble is that by canceling out an x from the numerator and denominator, you are effectively dividing both of them by 1/x, which is undefined with x is zero.
When you've identified all the values of x that cause problems for h(x), then you can say what the domain is (all other values).
Hope this helps,
Stephen La Rocque.