A Function Invariant under a Group of Transformations

The functions , , given by , , , , , and form a group with the composition of maps; for instance .

Moreover, if we take , we have , , that is, the function is invariant with respect to the group of transformations . The graph demonstrates this behaviour: choose a real number on the horizontal axis; the numbers , are shown on the same axis and the corresponding values (red points) are all at the same height.