How would you go about telling what the contour lines of a function such as min{x,y-1/x} has?

How, for example would you go about figuring out the kind of shape the graph would have?

I have graphed the function min{x, y-1/x} using wolfram but if am at a loss as to how I would figure out what it looks like without the aid of software.

A function such as min{ax,by} looks like a folded piece of paper along the line y=(b/a)x, but what, analytically speaking is the technique for gathering information on a general form of min{f(x),f(y)}

Also, what is this kind of problem called? If I googled min problems I generally get minimization problems.