Testing the Domain of a Function

function, domain, denominator

The domain of a function consists of all the possibly input (normally x) values. There are three possible scenarios that result in values not being in the domain:

1. When the denominator (if there is one) is equal to 0.

2. When taking the square root of a negative number

3. When taking the log of a non-positive (negative or 0) number.

Testing these three cases will normally eliminate all value that aren't in the domain. For example, take the function

log(10–x) / (x+4)

Testing the three cases:

The denominator ((x+4) ) is 0 when x = -4. Therefore, -4 is not in the domain.

There is a square root, and the value inside the square root is negative when x < -4. Therefore, x < -4 is not in the domain.

There is a log, and the value is non-positive when x10. Therefore x10 is not in the domain.

So, looking at these three values means the domain is -4 < x < 10.

A good visual technique while eliminating values for the domain is to use a number line and cross out the values that do not work.

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                        -4            0                         10                     

function, domain, denominator