The known analytical formula for composite functions is:
f[Sqrt[x] + 1/Sqrt[x]] == x + 1/x
The analytical formula for f [x] cannot be obtained through this method:
In[62]:= Clear["Global`*"]
RSolve[{f[Sqrt[x] + 1/Sqrt[x]] == x + 1/x}, f[x], x] // FullSimplify
Out[63]= RSolve[{1/x + x == f[(1 + x)/Sqrt[x]]}, f[x], x]
and this method is also cannot
Clear["Global`*"]
Simplify /@ (f[Sqrt[x] + 1/Sqrt[x]] == x + 1/x /.
Solve[y == Sqrt[x] + 1/Sqrt[x], x][[1]]) /. {y -> x,
Equal -> Rule}
Reduce[Sqrt[x] + 1/Sqrt[x] == x + 1/x]
is true when x=1 only. What isf
role here? I am not reading your question right it seems. $\endgroup$f
F represents the corresponding relationship of the function $\endgroup$Sqrt[x] + 1/Sqrt[x]
and want me to return backx + 1/x
then only whenx=1
this is possible. I am not sure what you mean by how to representf
in this case. Hopefully someone will have an answer. $\endgroup$