The correct answer is option 3: Epidermis. Root hairs are small, finger-like projections that extend from the epidermal cells of a plant`s root. These hairs increase the surface area of the root, allowing for greater absorption of water and nutrients from the soil. The epidermis is the outermost layer of cells in the root, and it is responsible for protecting the inner tissues of the root and controlling the movement of substances in and out of the root. While the cortex (option 1), pericycle (option 2), and endodermis (option 4) are all important layers of cells in the root, they do not directly give rise to root hairs. So, the correct answer is option 3: Epidermis.