# Domain, interval, subset of a function

Given the set of all real numbers, what's the subset of all real numbers that can server as a domain for this function:

$f(x) = \sqrt{x}$

I know that the answer is $x \ge 0$. But is it the same as writing [$0,+\infty$) ? Or same thing as Solution $= \{ x \in \mathbb{R} | x \ge \frac{5}{3} \}$ ?

• 5/3? Where did that 5/3 come from? – Gerry Myerson Oct 15 '11 at 22:25

If the question asks for the largest set of real numbers that can serve as the domain of the function $f(x)=\sqrt x$, it isn’t really correct to say that $x\ge 0$ is the answer: $x\ge 0$ is a condition on $x$, not a set of real numbers. The answer is $\{x\in\mathbb{R}:x\ge 0\}$ or $[0,\infty)$; both are correct, since they specify exactly the same set. Of course the same information can be expressed in many other ways. To give just one, one can say:
$\sqrt x$ is defined if and only if $x\ge 0$.
If the function were instead $g(x)=\sqrt{x^2-4}$, the answer could be given either in interval notation as $(-\infty,-2]\cup[2,\infty)$ or in set notation as $\{x\in \mathbb{R}:|x|\ge 2\}$; these specify exactly the same set and are equally correct. Yet another possibility is $\{x\in\mathbb{R}:x\le -2\}\cup\{x\in\mathbb{R}:x\ge 2\}$, mirroring the interval notation.
• Context is important. It is possible that some instructor has decreed that as far as Math 101 is concerned, $[0,\infty)$ is correct, and $\lbrace x\in{\bf R}:x\ge0\rbrace$ is forbidden (or the other way around). It's important to know what is mathematically correct, but it is also important (in a classroom situation) to know what the local rules are, as well. – Gerry Myerson Oct 15 '11 at 22:29
• @Gerry: Oh, absolutely, though I'd want to have words with such an instructor. It's also entirely possible that $x\ge 0$ would be perfectly acceptable in an elementary course. – Brian M. Scott Oct 15 '11 at 23:32