بِسْمِ اللَّهِ الرَّحْمَنِ الرَّحِيمِ — In the name of Allah, the Most Merciful, the Especially Merciful.
بِـ: preposition.
اسْمِ: genitive after bi-; mudāf.
اللَّهِ: genitive mudāf ilayh.
الرَّحْمَنِ، الرَّحِيمِ: definite genitive adjectives of Allah, both agreeing in case and definiteness.
Their function here is praise.