Which of the following is responsible for housekeeping functions?
Cox-1 are constitutive (always present in cells) , serve housekeeping function.
Cyclooxygenase  pathway:  All eicosanoids with ring structures (that is, the prostaglandins, thromboxanes, and prostacyclins) are synthesized via the cyclooxygenase pathway. Two related isoforms  of the cyclooxygenase enzymes have been described. ​
Cyclooxygenase-1 (COX-1) is responsible for the physiologic production of prostanoids, whereas cyclooxygenase-2 (COX-2) causes the elevated production of prostanoids that occurs in sites of chronic disease and inflammation. ​
COX-1 is a constitutive enzyme that regulates normal cellular processes, such as gastric cytoprotection, vascular homeostasis, platelet aggregation, and reproductive and kidney functions. COX-2 is constitutively expressed in tissues such as the brain, kidney, and bone. Its expression at other sites is increased during states of chronic inflammation.​
Differences in binding site shape have permitted the development of selective COX-2 inhibitors.​
Another distinguishing characteristic of  COX-2  is that its expression is induced by inflammatory mediators like TNF-α and IL-1 but can also be pharmacologically inhibited by glucocorticoids, which may contribute to the significant anti-inflammatory effects of these drugs.​
Reference: Lippincott Illustrated Reviews: Pharmacology, Wolters Kluwer Sixth Edition page no 448​
COX1
COX2
COX3
All of the above