A game engine is a set of functions that lets you draw to the screen, and other basic game functions like that. If you write your own engine, you will basically be writing functions that use DirectX or OpenGL and you will be calling those in your game. If you use somebody else's engine, then you will be calling functions that probably do the same thing, but you wont have written them. There is always the chance that the engine you are using will slow down performance of the game though.
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