If you're given a graph the range is all of the 
values and the domain is all of the 
values where the graph exists.
For example, consider this graph
What are the 
values at which the function is defined? Well we can see it starts at 
on the left and keeps going until 
. Notice, that even though this is a piecewise function, every single 
between 
and 
corresponds to a point on the graph. Then we just need to take into a account whether the endpoints are included or not. In this case 
is but 
is not. So the domain, in set builder notation, is 
.
As for the range, we look at the 
values. The lowest 
value at which the function is defined is 
. Then continuing up we see a break from 
to 
. There is no point on the graph that corresponds to 
values between those two numbers. But then it continues at 
and goes up to 
. In this case 
, 
, and 
are definitely included. It might be slightly harder to tell that 
is included, but it is. So the range is 
.
If you're given a graph the range is all of the values and the domain is all of the values where the graph exists.
For example, consider this graph
What are the values at which the function is defined? Well we can see it starts at on the left and keeps going until . Notice, that even though this is a piecewise function, every single between and corresponds to a point on the graph. Then we just need to take into a account whether the endpoints are included or not. In this case is but is not. So the domain, in set builder notation, is .
As for the range, we look at the values. The lowest value at which the function is defined is . Then continuing up we see a break from to . There is no point on the graph that corresponds to values between those two numbers. But then it continues at and goes up to . In this case , , and are definitely included. It might be slightly harder to tell that is included, but it is. So the range is .
I'm going to make some assumptions, let me know if this is what you have in mind.
Suppose we are given this graph:
and we are asked to give the domain and range of the function in set builder notation.
We can see that the function has a value at all points except where 
. So the domain of the function is 
Now we will consider the range. Note that the function goes off to 
near the origin, so most values are in the range of the function. However, this particular function never crosses the x-axis,
so 0 is not in the range. Therefore, the range is 
.
In all questions of this form, you have to first: identify the domain and range, and second: write it in set-builder notation. You can think about finding the range by imagining horizontal lines and seeing at what y-values they do (and do not) intersect the graph. Likewise with horizontal lines for the domain.
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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 
which has a domain constrained to some interval 
, then to write out the domain in this set-builder notation we would have 
, 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 
is clear if it is understood that the function we are discussing will always be real valued, for example. Furthermore, 
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 
we mean the set of all those & only those 
that are named, listed, specified, or defined by whatever "
" says. So 
regardless of what the set 
is. (But I suggest asking the students whether they know this!).And 
is accepted as meaning the same 
And using a comma instead of 
is acceptable here as it is not ambiguous or unclear. But 
is incomplete. E.g. 
and 
are different sets.
Let A be the set of all men and B the set of all women.
The cartesian product of A and B - AxB={(x,y): x∈A and y∈B}
Let R be a relation from A to B,
R={(x,y): (x,y)∈AxB and x is married to y)}
I know that the domain of R would be the set of all the initial components in the ordered pairs that belong to R and the range of R would be the set of all the final components in the ordered pairs that belong to R.
How can I denote the domain and range using the set builder notation?
and 
are on the graph in a), which means that your domain should actually be 
in interval notation to indicate that 
and 
are included. This would be reflected in the set-builder notation by using inequalities with "or equal to", so that you'd have 
.
- and 
-coordinates need to be included, so you'd need brackets for interval notation, and all 
's for the set-builder notation. So, for the range, I would write 
, using 
rather than 
.
for the domain. 
needs to only be "at most 
" (not "at least" anything), and thus you'll only need a single inequality, rather than the compound ones you would use on graphs that have a definite starting and ending point.