This seems very obvious and I am having a bit of trouble producing a formal proof.
sketch proof that the composition of two polynomials is a polynomial
be two complex polynomials of degree where and .
which is clearly a complex polynomial of degree .
sketch proof that the composition of two rational functions is a rational function
A rational function is a quotient of polynomials.
Notice that is a polynomial as
are both polynomials. Hence is a rational function as it is the quotient of polynomials.