Interval Notation

Date: 4/1/96 at 2:8:0
From: Maida Kelly
Subject: Interval Notation

Hello.  I have returned to college after many years and am taking 
a college level algebra course.  I need to learn about interval 
notation in terms of domain and ranges.  I would appreciate it if 
you could point me in the right direction as to where I could look 
up this information.  I can't tell which is harder, finding my way 
around the Internet or taking algebra.  Thank you for your time.

Please e-mail me at PtldLady@aol.com.

Thank you.

Date: 4/4/96 at 2:57:9
From: Doctor Aaron
Subject: Re: Interval Notation

Hello,

I think that what you want looks something like (a,b) or [a,b], 
where a and b are real numbers and a < b.  Let's take an example.  
Let a = 3 and b = 5.  Then we can talk about the interval [3,5]. 
This means all of the numbers between 3 and 5.  If we were just 
talking about integers, we could write the set {3,4,5}, but there 
are an infinite number of real numbers betweeen 3 and 5, so they 
wouldn't all fit in the set brackets, so we abbreviate with 
interval notation. We can use interval notation because any 
interval on the real number line is almost completely 
characterized by its endpoints.  The reason I say almost is 
because an interval may contain its endpoints and it may not 
contain its endpoints.  This is expressed in interval notation by 
a bracket or a parenthesis.

Here are some examples:

(3,5) is the set of all numbers greater than 3 and less than 5.

(2,4] is the set of all numbers greater than 2 and less than or 
equal to 4

[-1,1] is the set of all numbers greater than or equal to -1 and 
less than or equal to 1.

We call an interval of the form (a,b) open; [a,b] closed; 
[a,b) or (a,b] half-open or half-closed.

One objection that some people have when they are learning this 
notation is that open sets are not necessary because every 
interval must end somewhere.  In any finite set this is true, but 
we can get arbitrarily close to zero, or any other number, without 
actually getting there (you may have heard of Zeno's paradox).  
There is no actual number that is closer to zero than any other 
number.  If you have already thought about this it should not be 
surprising. If it's confusing don't feel bad; infinity is a weird 
thing.
 
I hope that this makes sense.

You also asked about domains and ranges of functions.

We think of a function that takes one number as input and spits 
out another number as output.  (Functions can work on things other 
than numbers, but it's usually best to think about functions of 
numbers first).  The most common way to represent a function is to 
let the x-axis be the input and the y-axis be the output.  Here 
are some examples of functions and domains:

The graph f(x) = 3 is the horizontal line where y is everywhere 3 
because if we pick any x as an input, f will spit back 3.  

The next question we want to ask is what x we are going to let f 
operate on.  We could choose (-3,5) or [-60000,12234). We have a 
lot of choices about which x we are going to let f operate on.  We 
say that the natural domain of f is (-infinity, infinity) because 
f will know what to do to any number that we could possibly think 
of - just turn it into 3.  We say that the set of numbers which f 
operates on is its domain.  The range of f is the set of outputs 
it produces. In this case the range of f is {3}

Now let's think about f(x) = 2*x

If we restrict the domain to (-3,5), then the range will be 
(-6,10) because if we allow f to operate on any number between 
-3 and 5, and we choose a number between -6 and 10, there will be 
a number (namely half of the number between 6 and 10) that is in 
(-3,5).  But the natural domain of f is really (-infinity, 
infinity) because there is no number that we can't multiply by 2.  
Then the range of f when its domain is (-infinity, infinity) is 
also (-infinity,infinity) because there is no number which we 
can't halve.  

Now let f(x) = 1/x here the natural domain is not 
(-infinity,infinity) because we can't divide by 0, its 
{(-infinity,infinity)-0}; another way to write this is: 
(-infinity,0) union (written upside-down U) (0, infinity) 

Now we can see why the distinction between open and closed 
intervals is useful.  We can also talk about the range of this 
function.  As x gets arbitrarily close to zero 1/x gets 
arbitrarily far away from zero, so we just have to look for holes 
in the range.  Well 1/x will never be zero, so the range is 
(-infinity,0) union (0, infinity).

One last example.  Let f(x) = x^2 + 5.  We can square any number, 
so the domain is the whole line.  However, any number squared is 
positive, so given an x, f(x) must be greater than or equal to 5. 
Then we say that the range is [5,infinity).

I hope that this is helpful.  

-Doctor Aaron,  The Math Forum