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A Function Invariant under a Group of Transformations

​
real number λ
3.35
The functions
H
i
(x)
,
i=0,1,…,5
, given by
H
0
(x)=x
,
H
1
(x)=
1
x
,
H
2
(x)=1-x
,
H
3
(x)=
1
1-x
,
H
4
(x)=
x-1
x
, and
H
5
(x)=
x
x-1
form a group with the composition of maps; for instance
H
5
(
H
2
(x))=
H
4
(x)
.
Moreover, if we take
f(x)=
3
(
2
x
-x+1)
2
2
x
(x-1)
, we have
f(
H
i
(x))=f(x)
,
i=0,1,…,5
, that is, the function
f(x)
is invariant with respect to the group of transformations
{
H
0
(x),
H
1
(x),…,
H
5
(x)}
. The graph demonstrates this behaviour: choose a real number
λ
on the horizontal axis; the numbers
H
i
(λ)
,
i=0,1,…,5
are shown on the same axis and the corresponding values
f(
H
i
(λ))
(red points) are all at the same height.
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