How do you know if your car thermostat is broken?

If the thermostat is broken in the closed position the engine would overheat rapidly. If it was broken in the open position the engine cooling system would not reach normal operating temperature and the heater would produce only warm air. Take the thermostat out and put it in a pan of water on the stove. When the water gets to around 200 deg F, the thermostat should open. If it doesn't it needs replaced. Observe carefully that you have not installed the thermostat upsidedown.