extend, give, cast, slave radical (no. 171)
隶 centers on the idea of reaching out or handing over: extending something, giving, or casting outward. The radical sense refers to this same shape used as a component in other characters, often related to slaves or servitude.
隶 is an ancient pictograph of a hand holding something, possibly a tail or a brush, conveying the idea of reaching or extending. Its exact historical development is uncertain, but it later became associated with servitude in compound characters.
The shape looks like a hand reaching out from a sleeve, extending to give or cast something. The lower part suggests a flowing motion, like water being poured out.
For タイ, imagine a hand extending to tie a rope: tie -> タイ, and the hand casts the rope outward.