Mina Y. answered 11/05/20
Bachelor's in Korean, Master's in Speech Therapy with +20 yrs teaching
"Alex is lost. We must go to find her."
For this sentence, I would use "Alex가 없어졌어요. Alex 찾으러 가야해요. "
.There are couple of ways to link two verbs in Korean. In Korean, for the meaning of "to do something", you can use -기 위해, -도록" like you said. But when you use those, it doesn't sound natural, is it? Because It is more like written style. But you can use simple connecting suffix "-러" which has meaning of "to do something" like below.
"Alex 찾으러 가야 해요." We must go to find Alex.
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"I would like to invite professor Kim to teach English"
For this sentence, I would use "김 교수님께서 영어를 가르쳐주시도록 초대하고 싶습니다."
When I see your sentence ending "싶습니다", I can tell that it is very polite sentence style. That's why I used honorific subject mark "-께서" and honorific suffix "-시" for verb (가르치다). And in this case, "-도록" is good and proper because it is full phrase has meaning of "for professor Kim to teach English." Professor Kim is the one who is going to teach, right? So you need make a phrase of "김 교수님께서 영어를 가르쳐주시도록".
Then, when can you use "-러"? When a person is the one to do both two actions eventually, you can use "-러" to link two verbs. Like "Alex 찾으러 가야해요." You are the one to do two actions of "go" and "find".
I hope this helps!!
Thanks,
Mina Yeon