I am wondering if anybody can give me a little bit of help with the duality
(or symmetry) property of the Fourier transform.  my book states that it is:

X(t) <-> 2(pi)x(-w)

...And this is pretty much all the text has to say about it.

first is the usage of X(t) as appose to x(t) in the situation to denote the
fact that we are using duality?  My book never mentions why it is suddenly
used, and this seems to be the only time it is used.

Also, when can I use this property, when is it applicable or even useful?
They don't even say where it was derived from, making it that much harder to
figure out.

thanks for any light you might be able to shed.