We discuss one of the most important aspects of rational functions; the domain restrictions.

You can watch a lecture video on this here!

Recall in the section on algebra of functions we mentioned a ratio of functions already; specifically . At the time we mentioned that there was something exceptional about this specific algebraic combination that was different than when we were simply adding, subtracting, or multiplying functions; the issue of domain.

We generally assume the natural domain (ie the largest possible domain) for a rational function unless a domain is explicitly given. We have discussed domains for each of the core functions we’ve already covered, but rational functions have their own special (but somewhat obvious) additional restriction; the denominator cannot equal zero. This shouldn’t be a surprise because we have already established that a value is undefined if we are trying to divide by zero, so wherever the function in the denominator equals zero would yield an undefined output.

A note about the previous example. An astute student may notice that the numerator is factorable as well, and that one of the factors cancels out, leaving you with a simplified form of being . It is very important to remember that the domain restrictions occur before any simplification process! This means that, although the domain restriction for can be “simplified away” it is still a domain restriction of the original function!

1 : Determine the domain of the function .