The air injection system operates at cold start mostly... You usually hear that vacuum cleaner high pitched noise on a cold startup, thats the air pump operating. The pump is on the right side of the engine bay, under the plastic cover, right behind the headlight. There are a number of hoses and pipes, and two valves, one on top of each valve cover, with more pipes leading to the exhaust manifold.
Not sure what happened to yours, but it wouldnt be the first time ive seen it fail. If you have a decent service history with Lexus, not just car purchases, then it wouldnt hurt to contact Lexus and seek some kind of assistance.
2014 Lexus GS350 FSport
2007 Toyota Tundra SR5 5.7 TRD 4x4
2010 Toyota Sienna Limited
2005 Subaru WRX STi