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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 tex2html_wrap_inline168 whole numbers tex2html_wrap_inline170 . Then the function can accept any whole number, but if you try plugging in tex2html_wrap_inline172 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 tex2html_wrap_inline176 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, tex2html_wrap_inline180 , so at tex2html_wrap_inline182 the denominator is zero and the function is undefined.
  4. Are there any even roots in the function, like tex2html_wrap_inline184 or tex2html_wrap_inline186 ? 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, tex2html_wrap_inline190 is defined only for x > 0.


Vector Calculus
Wed Sep 17 14:50:13 MST 1997