Step 1: In a root's transverse section, the layers from outside to inside are epiblema, cortex, endodermis, pericycle and then the vascular cylinder.
Step 2: Root hairs are simply outward tubular extensions of individual epidermal cells of the epiblema, formed to increase the surface area for absorption. They are not produced by any deeper layer like cortex, endodermis or pericycle.
Step 3: Since epiblema is the layer directly in contact with the soil, it is the layer that gives rise to root hairs.