For the first three, +, -, ⋅ the domain is the intersection of the domain of 
and 
. That is, the only restriction on the domain is that both 
and 
must be defined.
For division we must in addition restrict to 
such that 
.
For example, let 
and 
.
The domain of 
is 
. The domain of 
is 
.
The intersection of these is 
.
The domain of 
, 
and 
would all be 
.
For
we must also insist that the denominator is not zero, so we can not have 
.
The domain of 
is 
.
In the previous example we were able to write out 
as 
and it is clear from this expression what the domain should be.
In general we must be careful that both 
and 
are defined. This fact can be obscured when writing out the expression for the combination.
For example, if 
and 
then the domain of 
is all reals, and the domain of 
is 
.
What about the domain of 
? For both 
and 
to be defined we need to take the intersection of their domains, which is 
.
If we write out
We might be tempeted to simplify this and say 
, but this would be misleading because then it would appear that the domain is all reals.
The domain is not all reals, it is all non-zero reals, so we should say:
To find the domain of 
consider what it would take for 
to make sense.
needs to be defined, or we can't do the first step. But then we want to apply 
to the output of 
, so we need to know that 
is in the domain of 
.
Let 
and let 
. The domain of 
is 
.
The domain of 
is also 
.
To find the domain of 
we know that we need 
for 
to be defined, but in addition we need that 
is in the domain of 
, so
Therefore, the domain of 
is 
.
Note that if you just tried to write out the expression for 
it is clear that we need 
, but it's not so clear that we also need 
.
What if we do the composition in the other order, 
? Check that the domain of 
is 
.
Here is an even more basic illustration of this issue. Let 
.
The domain of 
is 
. The range of 
is also 
.
So if we compose 
with itself the domain of 
is 
.
However, if we write out
we may be tempted to simplify this expression to:
Be careful though, this expression makes it appear that the domain is all reals, but 
is not defined for 
.
If we wish to simplify we must write:
Author: Christopher Cashen <nil >
Date: 2008/09/23 10:50:10