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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?

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2 Answers 2

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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.

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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.

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    $\begingroup$ The function $f(x) = x^2$ is not its own inverse. $\endgroup$
    – march
    Feb 25, 2017 at 4:55

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