Airbags are designed to inflate in life threatening situations. They are essentially an explosive device that you don't want exploding in foront of you if you don't need it.
Given that you're able to post this concern, I'd say the accident did not rise to the level of "life threatening". I'm not being flip, I'm making a clinical evaluation. If the accident was bad enough to threaten your life, an airbag failure would have sent you to the hospital at a minimum.
In addition, the air bag computer on the car constantly evaluates the sensors and airbags on the car. This self check function is very strict, and will illuminate an air bag icon (light) on the dash when there is any malfunction in the system.
You do need to check that this self-check function is working. A yellow icon (a small picture of a seated driver with an air bag) should ALWAYS light up in your instrument cluster when you start the car. The light stays illuminated while the self-check runs and turns off after about ten seconds. If the light does NOT appear after you start, the likelyhood is high that someone bypassed the airbag system.
If you bought the car new, it's highly unlikely anything is wrong with the airbag system. However, if the car was bought used, it's possible the car has been in a crash, and someone defeated the airbag system. Anytime you buy a used car, check the dash to make sure the airbag system runs a self-check at startup. If the self-check runs, the airbag system is good and the airbag will deploy WHEN NEEDED.
