Functions that are their own inverse. What are the functions that are their own inverse?
(thus functions where $ f(f(x)) = x $ for a large domain)
I always thought there were only 4:
$f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ f(x) = \frac {-1}{x} $ 
Later I heard about a fifth one   
$$f(x) = \ln\left(\frac {e^x+1}{e^x-1}\right) $$ (for $x > 0$)
This made me wonder are there more? What are conditions that apply to all these functions to get more, etc.

Technically Any functions satisfying $$f(x)=-x+a$$ for real numbers $a$ are their own inverses
