cannot mirror MacBook air display to Apple TV

There could be many reasons for this. What might shed some light on this is information about the external display or TV you are trying to mirror your laptop to, and how it is connected to the Apple TV. Some description of symptoms or error messages would enable someone to try to troubleshoot this. Does the Apple TV work for other functions, such as streaming? Does it have an internet connection? Have you set it up with your Apple ID?

To mirror your laptop to an Apple TV and external display, you also need bluetooth turned on for your devices and they need to be on the same network.