The *domain* of a relation is the set of all the xx values for which there exists at least one yy value according to that relation. The *range* is the set of all the yy values, which can be obtained using at least one xx value.

If the relation is of height to people, then the domain is all living people, while the range would be about 0,1 to 3 metres — no living person can have a height of 0m, and while strictly it's not impossible to be taller than 3 metres, no one alive is. An important aspect of this range is that it does not contain *all* the numbers between 0,1 and 3, but at most six billion of them (as many as there are people).

As another example, suppose xx and yy are real valued variables, and we have the relation y=2xy=2x. Then for *any* value of xx, there is a value of yy, so the domain of this relation is the whole set of real numbers. However, we know that no matter what value of xx we choose, 2x2x can never be less than or equal to 0. Hence the range of this function is all the real numbers strictly greater than zero.

These are two ways of writing the domain and range of a function, *set notation* and *interval notation*. Both notations are used in mathematics, so you should be familiar with each.

A set of certain xx values has the following form:

x
:
conditions
,
more
conditions
x
:
conditions
,
more
conditions

(5)We read this notation as “the set of all xx values where all the conditions are satisfied”. For example, the set of all positive real numbers can be written as {x:x∈R,x>0}{x:x∈R,x>0} which reads as “the set of all xx values where xx is a real number and is greater than zero”.

Here we write an interval in the form '*lower bracket, lower number, comma, upper number, upper bracket*'. We can use two types of brackets, square ones [;][;] or round ones (;)(;). A square bracket means including the number at the end of the interval whereas a round bracket means excluding the number at the end of the interval. It is important to note that this notation can only be used for all real numbers in an interval. It cannot be used to describe integers in an interval or rational numbers in an interval.

So if xx is a real number greater than 2 and less than or equal to 8, then xx is any number in the interval

It is obvious that 2 is the lower number and 8 the upper number. The round bracket means 'excluding 2', since xx is greater than 2, and the square bracket means 'including 8' as xx is less than or equal to 8.

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