At LAX and many other large airports they designate certain runways for landing or takeoffs Normally. Why is this? I think its inefficient use of available runways.
There are many constraints on runway usage, and sometimes it's not the runways themselves but the protected airspace for approaches that determines what "normal" is - as well as airport design, noise abatement, wind, aircraft type, taxiway configuration, departure routes in use, etc. It is typically more efficient (= less delay) to run all of one type of operation on a runway: you can launch more departures faster if you're not trying to land other aircraft in between them, so if there is a major departure push on, the tower will try to dedicate at least one runway to departures only. Same thing for arrivals: landing aircraft need to be spaced further apart if there are departures going out on the same runway, and if a departure can't roll as soon as expected (for example, if the last arrival doesn't clear the runway as quickly as possible), the next arrival may have to go around because the next departure is still sitting on the numbers. Having said that, unless the airport management puts specific restrictions on runway usage, ATC is free to shuffle the deck any way they see fit and land/depart from any appropriate runway if operational needs require it. Pilots also have the right to request a specific runway for THEIR operational needs, and ATC will accommodate the request even if it isn't exactly their current plan.
LAX generally lands on the outboard runways because the final approaches are further apart, and taxiing is less complex if the departures go off the inboard runways. That doesn't mean they never do anything else, but that's kind of the "normal" case.
At LAX there are 4 runways (24/6 left and right, 25/7 left and right). All runways can be used for both takeoff and landing.
At certain airports there are operational circumstances such that a given runway is used primarily for departures/arrivals to handle various traffic/ATC needs or noise abatement programs. For example, if there becomes a significant number of aircraft lined up on the taxiway awaiting takeoff, ATC may decide to put arriving aircraft on one runway and depart aircraft on the other (normally parallel) runway. Or if a specific program exist for operational considerations at that airport.
By the way, LAX's use of runways is not a good example of other major airport runway use programs. Because of noise issues and as a result of a couple of accidents, LAX has a more regimented runway use program.
I'm not aware of any U.S. major airport that designates any runway "exclusively" (i.e., without exception) for takeoff or landing.