Using an iMac as a monitor for a MacBook air

Hi Kat_1127,

Unfortunately, Apple killed off Target Display Mode, and Apple silicon Macs don't support using older iMacs as a second display. The same applies with recent Intel Macs as well.

(Apple silicon Macs can only run Big Sur and newer - one update after macOS Catalina, the last supported version for this.)

Use your iMac as a display with target display mode - Apple Support

+1, in short you cannot. The MBA is too new and the iMac would need to be a late 2009 27" model. Buy yourself an external display, they are far easier to setup, maintain, and don't require expensive cabling like Target Display Mode did. You can find external displays all over in many price ranges beginning at about $100.

As you note, 21.5" Late 2009 iMacs like the OP's iMac never supported Target Display Mode.

With respect to expensive cabling (for iMacs that did support it) …

Target Display Mode on a 27" Late 2009 or 27" Mid 2010 iMac does not require expensive cabling. Those iMacs did not have Thunderbolt ports, and so when they are operating in Target Display Mode, they only need Mini DisplayPort input. If Apple had not added restrictions on "the other Mac", the required cabling for one of today's "other Macs" (USB-C (DP) to Mini DisplayPort adapter cable) would be readily available and quite affordable.

Once iMacs got Thunderbolt ports, TDM required Thunderbolt input, which made cabling more expensive - and eliminated the ability to use an iMac operating in TDM as a display for an older, non-Thunderbolt-equipped Mac. All of the 2011, 2012, 2013, and mid 2014 iMacs that support TDM suffer from this disadvantage.