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Last edited 4/14/2009 10:20:00 AM
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DOMAIN
AND CODOMAIN
A function has
associated with it two sets, its domain
and its codomain
. By
the specification
of function, we know that for a function
,
¨
(a)
is defined for every element
and for no other a.
¨
For every ,
(a)
must be an element of
. It is OK for there to be elements of
that are not
(a)
for any
.
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.
¨
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
value 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.
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
.
¨
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.