I've watched several videos, but they seemed to be doing different ways to write it. For example some domain would be; D [ 0, ∞] and some will be; Domain: { xER | x ≠ 0} Can anyone explain what's the difference between the two?
Videos
You can write the domain as 
, which is the more "normal" convention and the range 
or 
or even 
.
I would write 
for the domain. For the image notations vary; I would refer to the range as the image and write 
Alternatively, many people prefer notation such as 
or 
for the range.
Note that the way you have written the range is incorrect only because you haven't written the intervals correctly. What I presume you meant is 
which is correct.
I'm not going to claim to be a great authority on this question, but I think judging whether or not these answers are 'correct' depends on what you, as a teacher, are assessing. Generally speaking, the style of specifying the domain is done via set builder notation; where you create a set from some parent set, by first specifying elements of the parent set, and then what logical condition they must satisfy.
For example, if we had some real-valued function $f$ which has a domain constrained to some interval $[a,b]$, then to write out the domain in this set-builder notation we would have $\mathrm{Dom}(f) = \{x\in \mathbb{R} \mid a\leq x \leq b\}$, which would largely be accepted by most people to be clear and unambiguous.
So technically speaking, the answers they have provided does not conform to this traditional style. However, from reading what they have done, it is still very clear what they mean.
Sets like $\{x\mid 4\leq x \leq 7\} $is clear if it is understood that the function we are discussing will always be real valued, for example. Furthermore, $\{x\mid x\in\mathbb{R}, 4\leq x \leq 7\}$ is as clear as the usual set builder notation, but not exactly the same syntax.
So as a teacher, it's your choice to judge whether being precise/conforming to notation is more important than being clear and vice versa.
When we write $\{x|....\}$ we mean the set of all those & only those $x$ that are named, listed, specified, or defined by whatever "$....$" says. So $R=\{x|x\in R\}$ regardless of what the set $R$ is. (But I suggest asking the students whether they know this!).And $\{x\in R|....\}$ is accepted as meaning the same $\{x|x\in R\land ....\}.$ And using a comma instead of $\land$ is acceptable here as it is not ambiguous or unclear. But $\{x|4\le x\le 7\}$ is incomplete. E.g. $\{x|x\in\Bbb R \land 4\le x\le 7\}$ and $\{x|x\in\Bbb Q\land4\le x\le 7\}$ are different sets.
Notations vary. You should pick something straightforward, and state it clearly up front. One common choice would be $$\operatorname{dom} f\\ \operatorname{ran} f$$
another reasonable choice might be $$\mathscr{D}(f)\\\mathscr{R}(f)$$
Principia Mathematica used:
Unfortunately the term range means different things for different people.
In a function $f: \cal{D} \to \cal{R}$, we call $\cal{R}$ the codomain of $f$; it's the set where $f$ takes its values.
The image of $f$ is the set of values of $f$; it's a subset of the codomain, but usually smaller.
The term range means either codomain or image, and so is better avoided.
If you need notation, you may use $\text{dom}(f)$, $\text{codom}(f)$, $\text{im}(f)$.
Hey all, I’ve got a test tomorrow on this and I’m confused with the contingencies...
From what I’ve (tried to) closely paid attention to in class this is an example of how D and R are written:
D = { X € R / 1 < (greater or equal to) X < 10} R = { Y € R / 0 < Y < 10}
From what I understand one of the contingencies is when the bubble is filled in it means there can be many values (1.2, 1.3 etc.) but when it is not filled in it means it’s just one strict value or number.
So when it is a strict value it is written without the greater than signs ?
D = { X € R / 1 < X < 4} E = { Y € R / Y = 3}
Also we haven’t done function notation yet so that won’t be needed at the moment. But I’m seeing f(x) being used when do you use this?
Some help here....
Thanks!