The non-membranous organelles composed of RNA and proteins that function to assemble proteins are ribosomes.

Here's why:

* Non-membranous: Ribosomes are not enclosed by a membrane, unlike organelles like mitochondria or the Golgi apparatus.

* RNA and proteins: Ribosomes are made up of ribosomal RNA (rRNA) and proteins.

* Protein assembly: Ribosomes are the sites of protein synthesis. They read the genetic code from messenger RNA (mRNA) and use that information to link amino acids together, forming polypeptide chains which make up proteins.

Let me know if you'd like more information about ribosomes!