Domain

The domain of a function, f, can be thought of as the set of all numbers that can legally be plugged into the function. Thus, if the domain is the set whole numbers . Then the function can accept any whole number, but if you try plugging in you never know what will happen. In essence, the function doesn't know what to do with objects that are not in the domain.

To find the domain, we look for a few things:

1. Is the function explicitly not defined for some values of x? For example, in the function we know that we can only plug in real numbers that are greater than or equal to 2. The most common examples of this are the inverse trigonometric functions.
2. Is the function a rational function? This means that the function is a polynomial divided by another polynomial. Since polynomials can have zeros, if the polynomial on the bottom is zero, we are dividing by zero, which is not allowed. Any values of x which would imply division by zero, are implicitly outside the domain.
3. More generally, does the function have any function on the bottom which can go to zero? If so, any values which make the denominator zero are not included in the domain. For example, , so at the denominator is zero and the function is undefined.
4. Are there any even roots in the function, like or ? Since functions are usually only real valued (especially in this course) we want to avoid values of x that make the resulting function complex or purely imaginary.
5. Is there any other way the function is undefined? For example, is defined only for x > 0.