Last edited 3/13/2007 8:51:00 PM
DOMAIN
AND CODOMAIN
¨
If 
is a function with domain A, then (a)
is defined for every element and for no other a.
¨
If has codomain B, then for every ,
(a)
must be an element of B. It is OK for there to be elements of B that are not (a)
for any .
Example
If we say (where F
is the finite function defined above), then F(7) is not defined because .
And note that but there is no number a for which F(a) = 42.
Example
¨
If we say is defined by ,
then we can’t talk about G(3 + 2i), where 3 + 2i is a complex
number, because the domain of G
includes by definition only the real numbers.
¨
The formula is a perfectly well defined complex number,
namely 16 + 16i, but 3 + 2i
is nevertheless not in the domain of
G.
¨
And 3 is a real number, so it is in the codomain
of G, but the minimum v alue for G
is 4, so there is no real number a
for which . G(i 1) = 3, but i 1 is not in the domain of G.
A
function is not the same thing as its defining expression.
Example
Let be defined by . Then and ,
and all these
things are true:
¨
¨
¨ even though and . because the notation tells you that the domain of K
is .
Usage for domain and codomain
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Especially for functions defined by a formula on
the real numbers, texts often don’t specify the domain of a function. In that case the domain is usually regarded
as the set of all real numbers for which the formula is defined. For example, for G defined by ,
the domain is but for the function H defined by the domain is (the set of all real numbers except 0).
¨
If you have two functions in
which all the data is the same except which set is specified for the codomain,
are they the same function?
Mathematicians disagree on this, with the result that there are two definitions
of function in common use (more about that here.) Those who go
with the stricter definition say that if then and
are two different functions. Those who go with the looser definition say
they are the same.