Determine if the function's independent variables are real or complex. A function can have a complex domain even if the co-domain is real, and vice versa. For example, Euler's Identity uses the complex number i and results in a real number (-1).
Find the values for which the function is undefined. In the previously mentioned case, the function y = 1/x is undefined at x = 0, because dividing by zero is undefined. Another example is the function g = log h, which has a domain of the positive real numbers, because the limit of g as h approaches zero from the right is negative infinity, and log h is undefined for all h less than zero.
Find the values for which the limit of the function is infinity or negative infinity. Again, the limit of the function g = log h is negative infinity as h approaches 0 from the right, so zero cannot be in the domain of g. Similarly, the limit of the function y = 1/x approaches positive infinity as x approaches zero from the right and negative infinity as x approaches zero from the left, and thus zero is not in the domain of y.
Determine if the function is continuous. Attempt to create a "divide-by-zero" scenario by manipulating the function's form algebraically. If you cannot create such a scenario, then if p is the lowest value in the domain and q is the greatest value in the domain, all values between p and q are members of the domain.
Determine if the function is non-continuous, using the procedure in the previous step. If this is the case, then there exists at least one value that is between two consecutive members of the domain that is not itself an element in the domain (e.g., 3 is between 2 and 4, but 3 is not in the set of even integers).
Define the domain of the function as the set of all values excluding the values you identified above for which the function is undefined. For example, zero is not in the domain of y = 1/x, so the domain is defined as the set of real numbers excluding zero.