While driving, how does avoiding hard braking improve fuel efficiency?

dp questioned:

It is run of the mill these days to see news tales about how to save fuel by unreliable your driving habbits. One of the main ways is to “Get out of hasty starts and stops.” I can see why hasty starts demand more energy, but how does stopping the car vigorously also use more energy?

Whether the car’s momentum is lost owing to friction with the air and engine, or owing to friction with the brake pads, I can’t see how it would make any difference.

hows a car engine work in regard to fuel efficiency?

Rich questioned:

i watched a TV show days gone by where a driving lecturer was instruction how to get the best fuel efficiency out of a car. he confirmed that if your foot isn’t on the gas, then the injectors will be switched off and no fuel will be used? but if a car is idling it is subdue burning fuel right? so does the engine know the difference between idling when the car isn’t tender and when the car is in gear but tender below momentum and not life obsessed by its engine?