A self inverse function is a function $f$, such that $y=f(x)$, with the special property that $ff(x)=x$, or written another way, $f(x) = f^-1(x)$
Example:
Imagine there's a function $f$, such that $y = 1/x$
$f^-1(x) = 1/x$, which means that $f(x) = f^-1(x)$, therefore this specific function is said to be a self-inverse function.
Another example:
Let the function $f$ be such that $y = 3-x$
$ff(x) = 3-x$, { so, $ff(x) = x$ } therefore this specific function is a self-inverse function.
What is/are the domain(s) of these types of functions?
$\endgroup$ 21 Answer
$\begingroup$Such self-inverse functions are called involutions. Since the function $f : X \to Y$ and its inverse $f^{-1} : Y \to X$ coincide for an involution, the domain and codomain must be the same $X = Y$. Moreover, since $f$ must be a bijection (in order to have an inverse), the range must equal the codomain. So in the end, the domain of an involution $f$ must equal its range $f(X) = X$.
Other than that, the domain of an involution depends on its definition. For example, each of the following is a valid definition of an involution on the respective domain:
- $f : \mathbb{R} \to \mathbb{R} \quad f(x) = x$
- $f : [-1,1] \cap \mathbb{Q} \to [-1,1] \cap \mathbb{Q} \quad f(x) = -x$
- $f : \mathbb{R} \setminus \{0\} \to \mathbb{R} \setminus \{0\} \quad f(x) = \frac{1}{x}$
