What's the difference between "$[$" and "$)$" in "$f : [2, \infty) \to \mathbb{R}$"? I don't understand the difference between  "$[$" and "$)$" in defining the function like this:
$$f : [2, \infty) \to \mathbb{R} \qquad\text{and}\qquad f: x \mapsto \sqrt{x} − 2$$
Can anybody please help me understand them?
 A: The notation $f:A \to B$ means that $A$ is the domain of the function-- the set of input elements that the function maps, and $B$ is the co-domain-- the set of potential output elements that the function maps to.
Here the domain is the set $[2,\infty)= \{$all real numbers that are greater or equal to $2\}$.  The "$[$"" has nothing to do with defining the function but instead with defining the set.  $[a,b]$ is a closed interval.  It is the set of all real numbers between $a$ and $b$ including $a$ and $b$.  $(a,b)$ is an open interval.  It is the set of all real numbers between $a$ and $b$ not including $a$ and $b$.
And you can mixe the terms.
$(a,b) = $ the set of all possible numbers $x$ so that $a < x < b$.
$[a,b) = $ the set of all possible numbers $x$ so that $a \le x < b$.
$[a,b] = $ the set of all possible numbers $x$ so that $a \le x \le b$.
$(a,b] = $ the set of all possible numbers $x$ so that $a < x \le b$.
It is not to be confused with and has nothing whatsoever to do with the notation $f(x) =\sqrt{x}-2$ where the paranthesis mean.... we, they don't mean anything; they are just a place holder to indicate where the input of the function is.  This means you toss in the number $x$ and what you get out is the value $\sqrt{x} -2$.
In this excercise, that notation is being avoided entirely in favor of the "map to" notation.  $f: x \mapsto \sqrt{x} -2$ means that each $x$ is mapped to $\sqrt{x} - 2$.
Although you are probably used to seeing functions defined simply by some formula such as $f(x) = \sqrt x - 2$ in actuality a function must have other parts specified:
You must somehow specify the domain and co-domain, and somehow (to define it) the mapping of all pairs.  This notation does both.
$f: [2,\infty)\to \mathbb R$ means the function takes $[2,\infty) = \{x \in \mathbb R| x \ge 2\}$ as the domain and $\mathbb R$ as the codomain (although not every value of $\mathbb R$ will actually be mapped to).
And $f: x \mapsto \sqrt x - 2$ means each $x$ from $[2,\infty)$ will be mapped specifically to $\sqrt x - 2 \in \mathbb R$.
A: This is a standard notation for intervals. It helps define subsets of $\mathbb{R}$, which can then be used for setting as the domain (or codomain) of a function.
The set $[a, b]$ refers to all $x \in \mathbb{R}$ such that $a \le x \le b$. Note, in particular, that $x = a$ and $x = b$ satisfy this condition. So, the interval includes both its endpoints.
If you wish to exclude an endpoint, say $b$, replace the bracket $]$ with a parenthesis $)$. This notates a strict inequality $<$ instead of the non-strict inequality $\le$. For example, $[a, b)$ refers to the set of $x \in \mathbb{R}$ such that $a \le x < b$. Note that $a$ is still in this set, but $b$ is not.
Of course, you can similarly have $(a, b]$ (including $b$, but not $a$), or $(a, b)$ which includes neither $a$ nor $b$.
If you want an unbounded interval, that is, where the interval is unbounded in the positive and/or negative direction, we end our intervals with "$(-\infty,$" or "$,\infty)$". For example, $(a, \infty)$ refers to the set of points $x \in \mathbb{R}$ such that $a < x$ (if you like, the $< \infty$ is implied). If you wanted to include $a$, you'd write $[a, \infty)$.
Similarly, you can also write $(-\infty, b]$ for the set of all $x \in \mathbb{R}$ such that $x \le b$.
It is also valid to write $(-\infty, \infty)$, which is just equal to $\mathbb{R}$.
One final note: we don't tend to write brackets $[$ or $]$ when referring to $\pm \infty$. The reason is because the brackets suggest, in terms of notation, that $\pm \infty$ is included in the set, but there are no such real numbers. However, it is accepted in certain circumstances when dealing with the extended real numbers, but that's a whole different situation.
