Step 1: Understanding the Concept:
In coordination compounds where both cation and anion are complexes, we name the cation first followed by the anion. The oxidation state of the metal is indicated in Roman numerals.
Step 2: Detailed Explanation:
1. Cation: \([Ag(NH_3)_2]^+\). The ligand is ammonia, named as 'ammine' (with double 'm'). Two ligands make it 'diammine'. The metal is silver with oxidation state +1. Name: diamminesilver(I).
2. Anion: \([Ag(CN)_2]^-\). The ligand is cyanide, named as 'cyanido' in current IUPAC (or 'cyano'). Since it's an anion, the metal name ends in '-ate'. Silver becomes 'argentate'. Oxidation state is +1. Name: dicyanidoargentate(I).
3. Combined Name: diamminesilver(I) dicyanidoargentate(I).
Step 3: Final Answer:
The IUPAC name is diamminesilver(I) dicyanidoargentate(I).