1. publish or make something widely known
put out an appeal/statement
If you ‘put out’ something- such as an album, a report or a paper it means to make it available to the general public. You can use this expression in academic or professional contexts.
2. extinguish something
put out a fire/cigarette
If you ‘put something out’, such as a rumour or a fire it means to extinguish it or to eradicate it. This is a popular phrasal verb used in social and professional contexts.
3. irritate, upset, or inconvenience someone
I hope I haven't put you out
To ‘put someone out’ means to annoy someone or to inconvenience them in some way. You might say “I’d love to come and stay with you but I don’t want to put you out.” This is a less formal expression used more on social situations.