The domain of a function is the complete set of values for the independent variable that makes the function work. By 'work', we mean that the function can use these values as inputs and produce a result that is a real number.
As an example, consider the function This returns the square root of whatever we give it in x. If we gave it 16, it would return 4, the square root of 16. It would work with zero, since the square root of zero is zero.
But what if we gave it -4? There is no number that when multiplied by itself give a negative result* Therefore we say that negative numbers are not in the domain of the function.
So the domain of the function is
Domains can be written out in words as above, but to be more mathematically precise they are also written using either inequalities, or in interval notation. Considering again the function
As an inequality, we would write
Read as "the domain of the function is all values of x which are greater than or equal to zero".
For more on inequalities see Inequalities.
In so-called interval notation, the same function has a domain of This describes the set of values from 0 to positive infinity. The square brackets means the set includes zero and infinity themselves.