-기(가) is linked to an adjective that follows it and is used to evaluate/access the act that precedes it.
휴대폰이 없으면 친구들과 연락하기 힘들어요. or 이 카페는 조용해서 책 읽기 좋아요.
Adjectives that can follow are limited to ones that indicate evaluation or judgement, such as 쉽다, 어렵다, 좋다, 나쁘다, 편하다, 불편하다, or 힘들다.
Indicates that there is an intention or an idea to do something.
수업 후에 백화점에 갈까 하는데 같이 갈래요? or 영화 보기 전에 밥 먼저 먹을까 해요.
It can’t be used for things that have been completely decided.
Used to estimate or judge what an object is like.
기분이 나빠 보여요. 무슨 일 있어요? or 두 사람이 참 행복해 보이네요. or 유카 씨는 나이보다 훨씬 어려 보여요.
It is mainly used to indicate the reason or cause of something not being done or a negative result occurring.
축구 경기를 보느라고 숙제를 못 햇어요. or 음악을 듣느라고 전화 소리를 못 들었어요.
The subject of the preceding and subsequent clauses must be the same. It is used with verbs that take an amount of time. The subsequent actions and conditions must coincide with the preceding actions.