Final answer:
The medulla oblongata, part of the brainstem, contains the nuclei for cranial nerves IX, X, and XII, playing a vital role in controlling various autonomic and motor functions.
Step-by-step explanation:
The part of the brainstem that contains the nuclei for cranial nerves IX (glossopharyngeal nerve), X (vagus nerve), and XII (hypoglossal nerve) is the medulla oblongata. The medulla is the lowermost part of the brainstem, connecting the brain to the spinal cord. It houses important control centers for the cardiovascular and respiratory systems, as well as nuclei for the aforementioned cranial nerves which are responsible for various functions such as swallowing (CN IX), visceral control (CN X), and tongue movements (CN XII).