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c.1300, "piece of work imposed as a duty," from Old North French tasque (13c., Old French tasche, Modern French tâche) "duty, tax," from Vulgar Latin *tasca "a duty, assessment," metathesis of Medieval Latin taxa, a back-formation of Latin taxare "to evaluate, estimate, assess" (see tax). General sense of "any piece of work that has to be done" is first recorded 1590s. Phrase take one to task (1680s) preserves the sense that is closer to tax.
German tasche "pocket" is from the same Vulgar Latin source (via Old High German tasca), with presumable sense evolution from "amount of work imposed by some authority," to "payment for that work," to "wages," to "pocket into which money is put," to "any pocket."
"to put a strain upon," 1590s, from task (n.). Related: Tasked; tasking.