Let Xbe a setW set of extended-real valued functions on Xand 0 an element of Xsuch that w(0) = 0 for all win W. We give a general theory of dual representations, without any extra parameters, of various hulls for extended-real valued functions on X satisfying f(0)= inf f(X\{0})We use the unifying framework of quasi-convex functions with respect to families of subsets of Xand a condition generalizing the bipolar theorem. Our results contain, as particular cases, some recent results of Thach (1991, 1993) and Rubinov and Glover (1996) on W-pseudo-affine and W-quasi-coaffine hulls.We also give some results in the converse direction. These yield, in particular, that the bipolar theorem is equivalent to a certain property of the lower semi-continuous quasi-convex hulls of functions on locally convex spaces