late 14c., "action of making aromatic smoke as part of a ceremony," from Latin fumigationem (nominative fumigatio) "a smoking," noun of action from past-participle stem of fumigare "to smoke," from fumus "smoke, fume" (from PIE root *dheu- (1) "dust, vapor, smoke") + root of agere "to set in motion, to do, perform" (from PIE root *ag- "to drive, draw out or forth, move"). Sense of "exposure (of someone or something) to aromatic fumes" is c. 1400, originally as a medicinal or therapeutic treatment.

1844, "perform a function" (intransitive), from function (n.). Related: Functioned; functioning.

1530s, "one's proper work or purpose; power of acting in a specific proper way," from Middle French fonction (16c.) and directly from Latin functionem (nominative functio) "a performance, an execution," noun of action from funct-, past-participle stem of fungi "perform, execute, discharge," from PIE *bhung- "be of use, be used" (source also of Sanskrit bhunjate "to benefit, make benefit, atone," Armenian bowcanem "to feed," Old Irish bongaid "to break, harvest"), which is perhaps related to root *bhrug- "to enjoy." Meaning "official ceremony" is from 1630s, originally in church use. Use in mathematics probably was begun by Leibnitz (1692). In reference to computer operations, 1947.