Triggered by the interesting finding of Andrew I open this thread for the further investigation of elementary superfunctions, i.e. functions 
that are expressible with elementary functions and operations such that
=f(F(x)))
for a given elementary function
.
Our first example is:
=2x^2-1)
with a superfunction =\cos(2^x))
.
Now the
has the same property =2\cosh(x)^2 -1)
as the 
. Hence =\cosh(2^x))
is another superfunction of 
.
Indeed=F(t+F^{-1}(x)))
exists and is differentiable at 
.
But it does not exist at the other fixed point
, because )
is not defined.
Edit: both are regular super-functions at fixed point 1.=1)
.
So if we are at polynomials, we can also give an elementary superfunction for =x^a)
, i.e. =c^{a^x})
.
Because=c^{a^xa}=F(x)^a)
.
Edit: these are the regular super-exponentials at 1..
Generally for Chebyshev polynomials, these are the polynomials
such that =T_n(\cos(x)))
- for example above we used =2x^2-1)
-, we know already two elementary superfunctions of =T_n(x))
, these are =cos(n^x))
and =\cosh(n^x))
.
for a given elementary function
Our first example is:
Now the
Indeed
But it does not exist at the other fixed point
Edit: both are regular super-functions at fixed point 1.
So if we are at polynomials
Because
Edit: these are the regular super-exponentials at 1.
Generally for Chebyshev polynomials, these are the polynomials