The singer-songwriter shared the clip to IG early Sunday morning ... marking her first post since announcing her mother and sister passed away on the same day.

In it, MC sings "It's Like That" in the clip ... singing out the notes while a pianist accompanies her. She flashes a bright white smile at the camera during her mini performance.

Carey gets real about her feelings in the caption ... saying she's back at work after "a couple of rough weeks" -- thanking her fans for their constant support and adding she's excited to play for fans in China and Brazil very soon.

As you know ... Mariah's mother Patricia and sister Alison passed away on the same day about two weeks ago -- with Mariah telling TMZ she's happy she got to spend time with her before she passed, despite their complicated relationship.

When it comes to Alison, sources close to the situation told us she and Mariah weren't in contact before she passed ... with drugs playing a major factor in their estrangement.

The cause of death for her mom and sister is not yet known.

BTW ... nothing's been formally announced, but it looks like Mariah is indeed doing an abbreviated world tour this month -- this after fans suspected she'd be playing a few venues in Asia before coming back to the U.S. for her holiday tour.