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=xy=x diagonal line. It also satisfies the following properties:

f=f−1f = f^{-1}
f(f(x))=(f∘f)(x)=xf(f(x)) = (f\circ f)(x) = x

Well-known self-inverse functions include

x↦xx \mapsto x
x↦−x+kx \mapsto -x + k
x↦kxx \mapsto \frac kx
x↦xx−1x\mapsto \frac x{x-1}
x↦ax+bcx−ax\mapsto \frac {ax + b}{cx-a}

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