Is there a name for functions having this property :$f(f(x))$ = $x$

Question

Not all functions satisfy : $f (f(x)) =x $ for all $x$. .

For example, function $g$ such that $g(x)=x^2$ certainly don't.

For $g(g(x)=g(x^2)=(x^2)^2$=$x^4$ which is not equal to $x$ for all $x$, far from this.

Functions $h(x)=\frac {1} {x}$ ( provided $x$ is not 0) and $h(x)= -x$ do satisfy the property.

Has this property been given a name?

Answer 1

Functions with this property are called involutions or self-inverse functions.

When a function is regarded as a relation, the functions satisfying $f(f(x)) = x$ are exactly the functions which are symmetric relations. But they are (to my knowledge) never called "symmetric functions".

Answer 2

Functions with this property are known as involutions. Another example of an involution is the map $x \mapsto \frac{x}{x-1}$.