For F.B.I. agents, watching an Islamic State suspect in the United States is a study in anxiety. Being an Islamic State sympathizer is not against the law. Neither is expressing hatred for the United States on Twitter. Buying guns is also legal, and investigators have watched nervously as terrorism suspects passed background checks and purchased guns more than 2,000 times in the past decade, according to government data.

The threat from the Islamic State has put a unique demand on American counterterrorism officials. Nobody expects the F.B.I. to discover the angry, violence-obsessed young man and arrest him before he shoots his classmates. But the same person, inspired by the Islamic State, is a priority. Missing him is considered an intelligence failure.

So agents watch and wait, looking for some sign that yesterday’s angry man with a gun is about to become today’s terrorist. That is why investigators have moved to arrest people early, well before a plot is fully realized, and why agents often use undercover stings: They generate controversy but give the F.B.I. a measure of control.

The F.B.I. does not have explicit requirements on who should be monitored because of suspected ties to the Islamic State. But the bureau — whose agents constantly monitor chat rooms, the Islamic State’s Twitter accounts and other online traffic — focuses on people who have tried to directly contact the Islamic State through social media or have said they want to travel to Syria to join the group. The bureau also relies on information from relatives, friends, teachers, clergy members or others in the community about people suspected of ties to the Islamic State.