Inverse Quartic

This problem could fit in a good grade 11 class or in grade 12. An important notion in these grades is that of the inverse of a function.

The graph of the quartic (degree 4) polynomial

is drawn below.  Take note of the double roots at x = 1 and x = –1, and of the symmetry about the y-axis.

The inverse of this graph is drawn below at the right. It is the reflection of the original in the line y =x.  Points (a, b) in the original graph correspond to points (b, a) in the inverse.

Although  is a function, its inverse is not. However it can be broken into the four pieces shown in color at the right, such that each individual piece is the graph of a function.

 Your job is to find equations for each of these four functional pieces.