Finding the Domain of a Function

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When somebody gives us a function, one of the first questions we should ask ourselves is what numbers the function is defined on. The numbers the function is defined on is called the domain of the function. How can a function be “not defined” for some number? One problem occurs with division.

Remember, we can never divide by zero. So, if somebody hands us the function f(x)=\frac{1}{x}, we know that the domain can’t include zero since if we plug in zero for x, we get f(0)=\frac{1}{0}. Since we can’t divide by zero, f(0) is not defined, and therefore zero is not in the domain of f.

Similarly, if g(x)=\frac{1}{(x-1)(x-2)}, then 1 and 2 aren’t in the domain of g since plugging these in for x gives us a denominator of zero.

In both of these examples, we can plug in any other number without problem, so the domain of f is all real numbers except zero, and the domain of g is all real numbers except for 1 and 2.

Here are a couple of questions:

  1. What is the domain of h(x)=x^2+1?
  2. What is the domain of u(x) = \frac{1}{x^2-1}?
  3. What is the domain of v(x)=\frac{x-2}{x^2-x-2}?
  4. Think about your answer to the first question. Can you say anything about the domain of an arbitrary polynomial?