1^k as input for the adversary - why?

Hi,

I just recognized that in many definitions (e.g. for one way-
functions) the adversary gets as input 1^k where k is the key length.
I wonder why this is the case. In fact, I read in some lecture notes
that the reason for this is to rule out one way functions that are
only owfs because the adversary needs exponential time to print the
result of the inversion e.g. f^-1(f(x)). In fact if the adversary has
to ouput a value (i.e. write it on its output tape) that is
exponential long then its running time is not longer restricted by a
polynomial and thus we would have a owf, although maybe there is a
trivial strategy to invert the function. However, I don't see how
giving the adversary 1^k as input would eliminiate such functions. If
we give the adversary 1^k as input (exponential long) he has to read
it and thus would again run in exponential time. In this way every
function with a large enough domain would be a owf...

Probably I am missunderstanding something. Could you help me?

Thanks,
Theo

.