Why doesn't the range of $f(x) = |x|$ include $0$

Question

In my pre-calc book they show the range of $f(x) = |x$| as $[0,\infty)$. But why is this not inclusive of zero $(0,\infty)$? We have two cases with $f(x) = |x|$. They are $x < 0$ and $x\ge 0$. I get with $x < 0$ zero isn't included because x never equals 0 but for $x \ge 0$, $x$ can equal zero. And clearly $|0| = 0$ so what am I missing?

Answer 1

$$[0, \infty) = \{ x | x \ge 0\}$$

It is included in the range due to $f(0)=0$.

In contrast $$(0, \infty) = \{ x | x > 0\}$$ and hence

$$(0, \infty) \subset [0,\infty)$$

and the two sets differ by the element $0$.

Square braces include the boundary while round braces exclude the boundary.