And what about onto?
are "different" functions but they have the same Laplace transform.
For the 2nd question you really need to specify to co-domain.
Remember that s is a complex variable where as t is a real number
If L[f_1] = L[f_2] is it necessary for f_1 to be equal to f_2 ?
As my example shows if you change only a "few" points on a function it will not change its Laplace transform. Functions like this are considered the same if they only differ on a set of measure zero. Formally functions like this are considered the same function as they are in the same equivilence class. So in this sense they are equal.