Domain Rules

When you are asked to determine the domain of a function, you should go through each of the following steps.

It is possible to answer “yes” to more than one question.  In this case, you need to do the work described in all the questions that you answered “yes” to and combine the domain.

1.  Ask yourself, “Does the function have a denominator?”.  If the answer is yes, then you need to determine what number or numbers make the denominator equal to zero and eliminate those from the domain.  Remember that you cannot divide by zero.

Example:  Find the domain of  .

This function has a denominator.  We are not allowed to have zero in the denominator.  Thus we must find out what number will make the denominator zero.

Since   will make the denominator equal to zero, we must remove it from the domain.  The easiest way to write this is
This can be written in interval notation at .

2.  Ask yourself, “Does the function have a square root or even (4th ,6th, etc.) root in it?”.  If the answer is yes, then you need to determine what numbers will make the expression under square root negative and eliminate those numbers from the domain.  Remember that you cannot take the square root of a negative number and get a real number.  Usually the easiest method for doing this is to the take the expression under the square root (or even root), set it greater than or equal to zero, and then solve that inequality.

Example:  Find the domain of .

This function has a square root.  We cannot take the square root of a negative number and get a real answer.  Thus we must find out what number or numbers will make the expression under the square root negative.  In this case it is easier to just determine what numbers will make the expression under the square root zero or positive.

(Note that the sign switches when we multiply or divide by a negative number.)

Thus the domain is .
This can be written in interval notation as .

3.  Ask yourself, “Does the function have a log or ln in it?”.  If the answer is yes, then you need to determine what numbers will make the expression after the log or ln negative or zero and eliminate those numbers.  Remember that you cannot take the log or ln of a negative number or zero.  Usually the easiest method for doing this is to take the expression after the log or ln, set it greater than zero, and solve that inequality.

Example:  Find the domain of  .

This function has a log.  We cannot take the log of zero or a negative number.  Thus we must find out what number or numbers will make the expression after the log negative.  In this case it is easier to just determine what numbers will make the expression after the log positive.

(Note that the sign switches when we multiply or divide by a negative number.)

Thus the domain is .
This can be written in interval notation as .

4.  Ask yourself, “Is this a real-world problem?”.  If the answer is yes, then you need to determine what types of number do not make sense in the context of the problem and eliminate them.

If the answer to all of the above is no, then the domain is all real numbers.