Inverses of Real Power Functions

It is interesting and important to observe that a good graph of a monotonic function like u = x or u = x³ automatically provides a good graph of its inverse function. In the case of the identity function, the inverse is also the identity x = u, and in the case of the cubing function, the inverse is the cube root, u = x^1/3.

We obtain the graph of the inverse function by the simple device of rotating the graph about the diagonal of the square.

If we apply the same transformation to the graph of the squaring function we do not obtain a function graph since above the points of the positive u-axis we have two values for the square root. However the rotated graph of the squaring function does give the graph of the square root relation.