Self-inverse functions (HL)

A self-inverse function, or involution, means it is its own inverse function. Such a function is symmetric across the $y=x$ diagonal line. It also satisfies the following properties:

Well-known self-inverse functions include

For self-inverse rational functions in the form $x \mapsto \frac{ax + b}{cx + d}$, upon reflection across ${y = x}$ the horizontal and vertical asymptotes switch places. The horizontal asymptote is ${y = \frac ac = k}$ and vertical asymptote is ${x = -\frac dc = k}$. This means ${a = -d}$.