Let's say I have a function $f$ and it's inverse $f^{-1}$
Say I don't know how to define either explicitly but I know one thing, and that is that $f$ is it's own inverse.
$$f( f(x) ) = x$$
Mathematica is quite capable of manipulating expressions where functions are only referred to by name, but I wish it to also use the above information when doing so.
How do I define such to Mathematica?
Why not just this:
f[f[x_]] := x
Whenever f[f[something]] is encountered, it's replaced by something.
f[1]
(* f[1] *)
f[f[1]]
(* 1 *)
f[f[f[1]]]
(* f[1] *)
f[f[x_]] := x is a rule to replace f[f[x_]] by x whenever this pattern occurs.
Example:
f[x_] := x^2;
fInverse[y_] := Solve[f[x] == y, x]
fInverse[y]
{{x -> -Sqrt[y]}, {x -> Sqrt[y]}}
f[fInverse[x]]
(* {{(x -> 0)^2}, {(x -> 1)^2}} *)
Or
x /. f[fInverse[x]][[2, 1, 1]]
(* 1 *)
(There is no unique inverse when $x^2 = (-x)^2$.)
but you can try the above with f[x_]:= 3 x + 8, for example.
