Domain and Codomain

I had a nice chat this afternoon with one of my students. The first topic had to do with the notation f : D→S. Here is what we said:

hi prof walton, i have a quick question

Shoot (but don't hurt me.)

haha, ok so i am under the impression that when functions
say f: x → x that means that the domain and the codomain
are the same but the ranges can be different correct?

The codomain (after arrow) lists the type of numbers
that might be values in the output, while the domain
(before arrow) lists all of the numbers that are in the
list of potential inputs. The range is the list of numbers
that actually are outputs. Is this the question?

yep, so the range is a sub"field" of the codomain rite
it lists all possibles while the range is what numbers
are in the function ?

subset instead of sub"field". Otherwise, yes.

haha ya i was lookin for that word

The cheapest answer for codomain would be simply R
(all real numbers). If the codomain is listed as something
more specific, that helps us understand the function better,
but we still might skip some of the numbers in the set.

ic, so when im looking at f and g that have the same
codomain, that then would not imply that f(g) = g(f)
because the ranges could be different? or is
f(g) being = to g(f) relational to the domain only

Equality of functions requires that they have the same
domain and the same values at every point in that
domain. If the domain is different, then the functions
must not be equal. If the functions have different
values for any point, then the functions are not equal.

so d → s just means that the domain could produce
these outputs right?

That's right. The outputs must be somewhere in the
list known as S. And the inputs only make sense if they
are in D.

but s just means possible outputs because its the codomain rite

Yes, because it is the codomain (after arrow)

rite rite rite, gotcha, this stuff is weird