There are several "min" functions in Mathematica. In the "Numerical" category, there is a "Min" function that works on a numerical list. Then there's MinValue for finding the minimum of a whole function (FindMinimum) and other variations of this. What I'm looking for is a "min" function that takes several functions (f1, f2, f3, ...) and returns the output of function (fn) across the domain that it (fn) has the lowest values. Say,

Min(-x^2, -(x + 1)^2)
=> -x^2         when x < -1/2,
   -(x + 1)^2 when x >= -1/2

Is there such a function in Mathematica? If not, how do I define one? It has to seemlessly work with all the other built-in functions in Mathematica so I don't always have to be worrying about the gaps.