We theoretically study trapped one-dimensional Fermi gases in the presence of spin-orbit coupling induced by Raman lasers. The gas changes from a conventional (nontopological) superfluid to a topological superfluid as one increases the intensity of the Raman lasers above a critical chemical-potential-dependent value. Solving the Bogoliubov-de Gennes equations self-consistently, we calculate the density of states in real and momentum space at finite temperatures. We study Majorana fermions (MFs) which appear at the boundaries between topologically trivial and topologically nontrivial regions. We linearize the trap near the location of a MF, finding an analytic expression for the localized MF wave function and the gap between the MF state and other edge states. © 2012 American Physical Society.