Old English of, unstressed form of æf (prep., adv.) "away, away from," from Proto-Germanic *af (source also of Old Norse af, Old Frisian af, of "of," Dutch af "off, down," German ab "off, from, down"), from PIE root *apo- "off, away." Compare off (prep.).

The primary sense in Old English still was "away," but it shifted in Middle English with use of the word to translate Latin de, ex, and especially Old French de, which had come to be the substitute for the genitive case. "Of shares with another word of the same length, as, the evil glory of being accessory to more crimes against grammar than any other." [Fowler]

Also by 1837 of in print could be a non-standard or dialectal representation of have as pronounced in unstressed positions (could of, must of, etc.)

Old English æfter "behind; later in time" (adv.); "behind in place; later than in time; in pursuit, following with intent to overtake" (prep.), from of "off" (see off (adv.)) + -ter, a comparative suffix; thus the original meaning was "more away, farther off." Compare Old Norse eptir "after," Old Frisian efter, Dutch achter, Old High German aftar, Gothic aftra "behind;" also see aft. Cognate with Greek apotero "farther off," Old Persian apataram "further."

From c. 1300 as "in imitation of." As a conjunction, "subsequent to the time that," from late Old English. After hours "hours after regular working hours" is from 1814. Afterwit "wisdom that comes too late" is attested from c. 1500 but seems to have fallen from use. After you as an expression in yielding precedence is recorded by 1650.

"retreat, stop annoying someone," by 1938, from the verbal phrase, from back (v.) + off (adv.).

"drive (something) away by violent blows," 1640s, from beat (v.) + off (adv.). The meaning "masturbate" is recorded by 1960s.

"initial burst of energy that launches a rocket into space," 1950, from the verbal phrase; see blast (v.) + off (adv.).

1709, "put aside, rejected," from verbal phrase cast off "discard, reject" (c. 1400), from cast (v.) + off (adv.). From 1741 as a noun, "person or thing abandoned as worthless or useless." Related" Cast-offs.

also cutoff, 1640s, "act of cutting off," also "portion cut off," from verbal phrase cut off (see cut (v.) + off (adv.)). Sense of "new and shorter channel formed on a river" (especially the Mississippi) is from 1773; of road that cut off or shorten a route, from 1806; of clothing (adj.), from 1840. Cutoffs "jeans or other long pants trimmed down to be shorts" is by 1967.

The verbal phrase is attested from late 14c. as "detach by cutting;" from 1570s as "exclude from access" and "bring to an abrupt end;" and from 1590s as "intercept, stop the flow or passage of."

also faceoff, 1886 in sports (hockey, etc., originally lacrosse), from verbal phrase in a sports sense, attested from 1867 (see face (v.) + off (adv.)); the off perhaps is based on stand-off or similar constructions.

also faroff, "distant, remote," 1590s, from adverbial phrase, from far (adv.) + off (adv.).

"escape," c. 1600, from get (v.) + off (adv.). Sexual sense attested by 1973.