What is the function of the farrier's hammer?

The farrier's hammer is specifically designed for driving nails into the hoof when applying shoes. It plays a critical role in the shoeing process, allowing the farrier to securely attach the shoe to the horse's hoof by means of nails. This process requires precision, and the hammer is crafted to provide the necessary force to embed the nails without damaging the hoof or causing pain to the horse.

Each of the other options represents tasks that are not typically associated with the function of the hammer. Trimming a horse's mane involves scissors or clippers, checking hoof balance is generally performed with different tools to assess how the hoof sits on the ground, and supporting the hoof while shoeing is usually done with a hoof stand or the farrier's hands. Thus, the primary and most accurate function attributed to the farrier's hammer is indeed for driving nails into the hoof during the shoeing process.