Range of self-invertible real valued function

Let \( f:\mathbb{R}\longrightarrow\mathbb{R} \) be a function of the form \[ f(x) = \dfrac{ax+b}{cx+d}, \quad\text{where}\quad a,b,c,d \neq 0 \quad\text{and}\quad cx+d \neq 0. \] Further, assume that \[ f(19) = 19, \quad f(97) = 97, \quad f(f(x)) = x \quad\text{for all}\quad x \neq -\dfrac{d}{c}.\] Find the range of \(f\).