Summary: ``This research project is an investigation into quadratic rational maps, $\varphi$, of one complex variable that map the unit disk to itself. Previous research [7] [MR3635761] shows that for each $\varphi$, a corresponding linear fractional map $\zeta$ can be found using the coefficients of $\varphi$, and this $\zeta$ can be used to characterize functions in the kernel of the adjoint of the composition operator with symbol $\varphi$, defined on a space of analytic functions. In this paper, we show sufficient conditions to ensure that certain cases of $\varphi$ map the unit disk to itself and find all the forms of $\zeta$. Finally, for a specific $\varphi$, we search for the kernel of its corresponding composition operator by calculating the matrices of C$\varphi$ and of the Toeplitz operator ${\rm T}_{-z\zeta'(z)/\zeta(z)}$, defined by projecting $\frac{-z\zeta'(z)}{\zeta (z)} f(z)$ onto the space of analytic functions.''