There is a fundamental problem here: y= |x+ 4| is NOT 'one to one' and so does NOT have an inverse! In order to have 'inverses', we would neet to separate the function at x= -4. For [itex]x\ge -4[/itex], [itex]x+ 4\ge 0[/itex] so y= x+4. The inverse of that is, of course, y= x-4 defined only for [itex]x\ge 0[/itex]. For x< -4, x+ 4< 0 so |x+4|= -(x+4)= -x- 4 so y= -x- 4. The inverse of that is y= -x- 4 again. And that, also, is defined only for [itex]x\ge 0[/itex].