Final answer:
The free-living and photosynthetic generation of the liverwort Marchantia is the gametophyte, which is the dominant stage of its life cycle.
Step-by-step explanation:
The generation of the liverwort Marchantia that is free-living and photosynthetic is the gametophyte. This stage of the liverwort life cycle is dominant, more visible, and longer lasting than the sporophyte generation. The gametophyte is characterized by a flattened body, known as a thallus, which spreads across the ground and performs photosynthesis. Marchantia is known for its complex thalloid structure with multiple layers of cells, including a skin layer with pores for carbon dioxide entry, a layer of chlorophyll-containing cells for photosynthesis, a storage layer, and an epidermis to anchor the plant. Sex and reproduction occur on the gametophyte, where specialized structures called antheridiophores and archegoniophores grow to produce sperm and eggs, respectively.