i believe as the turkey does not feel any happiness it should be either 0 or undefined, resulting in the function being undefined beyond death either way.
True. If it realizes death is iminent, it spikes as this will, at some scale, still be a gradual process where happiness approaches zero before actually reaching it. If not, it stays wherever it would normally be in those instances. Starting at the moment of death, undefined.
Getting a free-range, heritage turkey from a local farm is well-worth it, if it is affordable (or you can pool money to make it affordable). You can genuinely taste a significant difference. Not only do they grow for a longer period and get a more varied diet but they have a less stressful, happier life with more experiences. I think that the happiness makes them taste better.