Rex was a physicist who worked for the DHARMA Initiative. He was seen at the barracks and in the Orchid Orientation film.
On the Island
1977
Rex was seen using the lawnmower near Pierre Chang, when he was going to the filming location to film the Arrow Orientation film. ("Because You Left")
When the new DHARMA member arrived the Island, Rex was seen helping them inside the Processing Center. ("Namaste")
Later, Rex was seen guarding the Galaga after Faraday and the other scientists had arrived on the Island. ("The Variable")
Later
Rex was last seen in the Orchid Orientation film, assisting Dr. Pierre Chang in the experiment with the rabbit. ("There's No Place Like Home, Part 3") However, it's unknown what happened to him afterward. He may have been killed along with other members of the DHARMA Initiative in the Purge, possibly defected to The Others prior to The Purge, or he may have left the island by some other means.