Math Insight

Domain of functions


A function $f$ is a procedure or process which converts input to output in some way. A traditional mathematics name for the input is argument, but this certainly is confusing when compared with ordinary English usage.

The collection of all ‘legal’ ‘reasonable’ or ‘sensible’ inputs is called the domain of the function. The collection of all possible outputs is the range. (Contrary to the impression some books might give, it can be very difficult to figure out all possible outputs!)

The question ‘What's the domain of this function?’ is usually not what it appears to be. For one thing, if we are being formal, then a function hasn't even been described if its domain hasn't been described!

What is really meant, usually, is something far less mysterious. The question usually really is ‘What numbers can be used as inputs to this function without anything bad happening?’.

For our purposes, ‘something bad happening’ just refers to one of

  • trying to take the square root of a negative number,
  • trying to take a logarithm of a negative number,
  • trying to divide by zero, or
  • trying to find arc-cosine or arc-sine of a number bigger than $1$ or less than $-1$.

Of course, dividing by zero is the worst of these, but as long as we insist that everything be real numbers (rather than complex numbers) we can't do the other things either.

For example, what is the domain of the function $$f(x)= \sqrt{x^2-1} \hbox {?}$$ Well, what could go wrong here? No division is indicated at all, so there is no risk of dividing by $0$. But we are taking a square root, so we must insist that $x^2-1\ge 0$ to avoid having complex numbers come up. That is, a preliminary description of the ‘domain’ of this function is that it is the set of real numbers $x$ so that $x^2-1\ge 0$.

But we can be clearer than this: we know how to solve such inequalities. Often it's simplest to see what to exclude rather than include: here we want to exclude from the domain any numbers $x$ so that $x^2-1<0$ from the domain.

We recognize that we can factor $$x^2-1=(x-1)(x+1)=(x-1)\;(x-(-1)).$$ This is negative exactly on the interval $(-1,1)$, so this is the interval we must prohibit in order to have just the domain of the function. That is, the domain is the union of two intervals: $$(-\infty,-1] \cup [1,+\infty)$$


  1. Find the domain of the function $$f(x)={ x-2 \over x^2+x-2 }.$$ That is, find the largest subset of the real line on which this formula can be evaluated meaningfully.
  2. Find the domain of the function $$f(x)={ x-2 \over \sqrt{x^2+x-2 }}.$$
  3. Find the domain of the function $$f(x)=\sqrt{x(x-1)(x+1)}.$$