In a normal root, root hairs arise at

Root cap is present around the root apical meristem where the cells are immature and go on dividing, to giver rise root cap towards the distal region and majority of cells towards the proximal region. The cells, which are cut towards the proximal side eventually elongate and form a part of elongation region in which the cells started to differentiate. Epidermal cells proximal to elongation region now show tubular outgrowths which are called as root hairs and from this region the maturation region starts and ends up at the base of the stem. Root hairs are formed to increase the area of absorption.