The Domain of a Function

The domain of a function is the list of all possible values for the independent variables. In other words, the domain is any and all input values which will produce an output. Usually, the domain of a function is from negative infinity to positive infinity, but some functions have restricted domains. Things to check to see if the function has a restricted domain will be if there is a fraction, a square root, or a logarithmic function.

When dealing with a fraction, it is important to remember that if the denominator equals 0, then the function is undefined. So, for every value of x which would result in the denominator equaling 0, is not in the domain.

The square root of a negative number is an imaginary number. So, every value of x which would produce a negative inside the square root would not be in the domain of the function.

A logarithmic function cannot have 0 or a negative value inside the logarithmic function. So, any x value which produces a 0 or a negative value than that x value is not in the domain.

There are two ways to write the domain of a function. We can either use parenthesis () or brackets []. If we use the parenthesis that means we are not including that specific value, but including the numbers up to it. While the brackets, on the other hand, indicates we are including that value.

For example, if we were to write the domain as (3, ∞) means that the domain of the function goes from 3 to infinity, but does not include the value of 3. We could have the value of 3.000000000000000000000000000000001, but not 3 exactly. We can get extremely close to 3, but we will never get to 3 exactly. If instead, we wrote the domain as [3, ∞) then we would include the value of 3. It is important to note that whenever the domain includes infinity it is always represented with a parenthesis and never a bracket.