It is wise to be concerned about radiation exposure, but the amount of radiation in a chest x-ray is very small.
Consider this: The unit of measure that is usually used to describe radiation exposure is the millisievert (mSv). In the United States, we are exposed to about 3 mSv every year, just from radiation naturally occurring in the atmosphere. By comparison, a chest x-ray delivers a radiation dose of about 0.1 mSv. That means that the radiation dose from a chest x-ray is the equivalent of about 10 days of exposure to ordinary background radiation.