For example: I need to do a Fourier Transform on f(t)*h(t), h(t) is 1 for t>0 and 0 for t <0, the answers for several questions end up using the usual formula but integrate from 0 to infinity which is fine, however should we not divide by 2 to account for the -infinity to 0 part of the function?
I have done many similar questions recently in learning the Fourier Series and when simplifying the integral in a similar fashion, we ALWAYS divided by two.
Example:
In previous cases where you have seen a factor of $2$ or $\tfrac12$, there must have been a reason for that – some symmetry perhaps, but it's impossible to say for sure, since you haven't given any examples. But the reason cannot have been only that the interval of integration was cut in half. You must also take into account what function you are integrating.
Anyway, in this case here, the transform is $$ \begin{split} \int_{-\infty}^\infty f(t) \, h(t) \, e^{-it\omega} dt &= \int_{-\infty}^0 f(t) \, h(t) \, e^{-it\omega} dt + \int_{0}^\infty f(t) \, h(t) \, e^{-it\omega} dt \\ & = \int_{-\infty}^0 f(t) \cdot 0 \cdot e^{-it\omega} dt + \int_{0}^\infty f(t) \cdot 1 \cdot e^{-it\omega} dt \\ & = 0 + \int_{0}^\infty f(t) \, e^{-it\omega} dt , \end{split} $$ so there is no division by two.