Jax was the operator of the Vigilant.
When Axel was captured by the Agents, Niobe, who was given order to make a full recall, was at the airport when she called him and he quickly informed her of the situation. If the Agents get in the air with Axel, Jax will have to kill him. But luckily, Niobe and Ghost were able to save him.
While the teams of the three ships were in the Matrix to help Neo go to the Source, Jax was with Axel on the console, when the alarm warned them of approaching Sentinels. Axel went to the bridge to observe an incoming Tow Bomb thrown by the Sentinels when they were out of EMP range, but unknown to him, the catwalk to the bridge was damaged by the weight of Axel's leg brace. When he went back to warn Jax, the catwalk gave way; Axel fell to his death while a piece of the catwalk impaled Jax from behind, killing him.