Etymology
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for- 

prefix usually meaning "away, opposite, completely," from Old English for-, indicating loss or destruction, but in other cases completion, and used as well with intensive or pejorative force, from Proto-Germanic *fur "before, in" (source also of Old Norse for-, Swedish för-, Dutch ver-, Old High German fir-, German ver-); from PIE *pr-, from root *per- (1) "forward," hence "in front of, before, toward, near, against."

In verbs the prefix denotes (a) intensive or completive action or process, or (b) action that miscarries, turns out for the worse, results in failure, or produces adverse or opposite results. In many verbs the prefix exhibits both meanings, and the verbs frequently have secondary and figurative meanings or are synonymous with the simplex. [Middle English Compendium]

Probably originally in Germanic with a sense of "forward, forth," but it spun out complex sense developments in the historical languages. Disused as a word-forming element in Modern English. Ultimately from the same root as fore (adv.). From its use in participles it came to be an intensive prefix of adjectives in Middle English (for example Chaucer's forblak "exceedingly black"), but all these now seem to be obsolete.

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for (prep.)
Old English for "before, in the sight of, in the presence of; as far as; during, before; on account of, for the sake of; in place of, instead of," from Proto-Germanic *fur "before; in" (source also of Old Saxon furi "before," Old Frisian for, Middle Dutch vore, Dutch voor "for, before;" German für "for;" Danish for "for," før "before;" Gothic faur "for," faura "before"), from PIE root *per- (1) "forward," hence "in front of, before," etc.

From late Old English as "in favor of." For and fore differentiated gradually in Middle English. For alone as a conjunction, "because, since, for the reason that; in order that" is from late Old English, probably a shortening of common Old English phrases such as for þon þy "therefore," literally "for the (reason) that."
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care (v.)

Old English carian, cearian "be anxious or solicitous; grieve; feel concern or interest," from Proto-Germanic *karo- "lament," hence "grief, care" (source also of Old Saxon karon "to lament, to care, to sorrow, complain," Old High German charon "complain, lament," Gothic karon "be anxious"), said to be from PIE root *gar- "cry out, call, scream" (source also of Irish gairm "shout, cry, call;" see garrulous).

If so, the prehistoric sense development is from "cry" to "lamentation" to "grief." A different sense evolution is represented in related Dutch karig "scanty, frugal," German karg "stingy, scanty." It is not considered to be related to Latin cura. Positive senses, such as "have an inclination" (1550s); "have fondness for" (1520s) seem to have developed later as mirrors to the earlier negative ones.

To not care as a negative dismissal is attested from mid-13c. Phrase couldn't care less is from 1946; could care less in the same sense (with an understood negative) is from 1955. Care also has figured since 1580s in many "similies of indifference" in the form don't care a _____, with the blank filled by fig, pin, button, cent, straw, rush, point, farthing, snap, etc., etc. Related: Cared; caring.

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care (n.)

Old English caru, cearu "sorrow, anxiety, grief," also "burdens of mind; serious mental attention," in late Old English also "concern, anxiety caused by apprehension of evil or the weight of many burdens," from Proto-Germanic *karō "lament; grief, care" (source also of Old Saxon kara "sorrow;" Old High German chara "wail, lament;" Gothic kara "sorrow, trouble, care;" German Karfreitag "Good Friday;" see care (v.)).

Meaning "charge, oversight, attention or heed with a view to safety or protection" is attested from c. 1400; this is the sense in care of in addressing (1840). Meaning "object or matter of concern" is from 1580s. To take care of "take in hand, do" is from 1580s; take care "be careful" also is from 1580s.

The primary sense is that of inward grief, and the word is not connected, either in sense or form, with L. cura, care, of which the primary sense is pains or trouble bestowed upon something. [Century Dictionary]
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care-worn (adj.)
also careworn, "oppressed or burdened with cares," 1828, from care (n.) + worn.
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go for (v.)
1550s, "be taken or regarded as," also "be in favor of," from go (v.) + for (adv.). Meaning "attack, assail" is from 1880. Go for broke is from 1951, American English colloquial.
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health-care (n.)
also healthcare, 1915, from health + care (n.).
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care-free (adj.)
also carefree, "free from cares," 1795, from care (n.) + free (adj.). In Old English and Middle English this idea was expressed by careless.
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after-care (n.)
"care given after a course of medical treatment," 1854, from after + care (n.).
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care-worm (n.)
a word listed in 2nd print edition OED, whose editors found it once, in the 1598 edition of W. Phillip's translation of John Huyghen van Linschoten's account of his voyage to the East Indies, and marked it "? error for EAREWORM." But care-worm could be a useful word.
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