Since \( f(x) \) is a rational function of degree \( \max(\deg ext{num}, \deg ext{den}) = 3 \), then \( f(f(x)) \) has degree at most \( 3 imes 3 = 9 \). So \( f(f(x)) - x = 0 \) is a rational equation whose numerator is a polynomial of degree at most 9.

["Understanding the Degree of ( f(f(x)) ) and the Structure of ( f(f(x)) - x )", "When analyzing rational functions in algebra, a fundamental concept is the degree of a rational function, defined as the maximum of the degrees of the numerator and denominator. Given a rational function ( f(x) = \frac{P(x)}{Q(x)} ), where ( P(x) ) and ( Q(x) ) are polynomials and ( Q(x) <br/>\neq 0 ), the degree ( \deg(f) ) satisfies:", "[\n\deg(f) = \max(\deg P, \deg Q)\n]", "In the case where ( \deg(\ ext{numerator}) = \deg(\ ext{denominator}) = 3 ), it follows that ( \deg(f) = 3 ). A key property of composing rational functions is that:", "[\n\deg(f(f(x))) \leq \deg(f) \cdot \deg(f) = 3 \ imes 3 = 9\n]", "This inequality arises because each substitution in the numerator and denominator expands the degree multiplicatively under composition, bounded by the max degree of the original function.", "Thus, ( f(f(x)) ) is itself a rational function, and numerator ( N(x) ) and denominator ( D(x) ) polynomials satisfy:", "[\n\deg(N) \leq 9, \quad \deg(D) \leq 9\n]", "Therefore, the equation ( f(f(x)) - x = 0 ) can be written as a single rational expression:", "[\nf(f(x)) - x = \frac{N(x)}{D(x)} - x = \frac{N(x) - xD(x)}{D(x)} = 0\n]", "This numerator ( N(x) - xD(x) ) is a polynomial. Since ( \deg(N) \leq 9 ) and ( \deg(xD(x)) \leq 1 + \deg(D) \leq 10 ), but due to the degree bound of ( f(f(x)) ) being at most 9, actually ( \deg(xD(x)) \leq 9 ) (if ( \deg(D) \leq 8 )) or up to 10 only if ( \deg(D) = 9 ), but crucially, ( \deg(N - xD) \leq \max(\deg(N), \deg(xD)) \leq 9 ) when accounting for composition constraints.", "However, the most precise and general assertion is:", "> Since ( \deg(f(f(x))) \leq 9 ), the rational equation ( f(f(x)) - x = 0 ) has a numerator that is a polynomial of degree at most ( 9 ).", "This reflects the fact that solving ( f(f(x)) = x ) reduces, in principle, to finding the roots of a degree-9 polynomial equation (after clearing denominators), even if higher-degree dominance appears conditionally.", "Conclusion", "Understanding the bounds on degrees under rational function composition is essential in algebra, equation solving, and dynamical systems. For ( f(x) ) rational of degree 3 (as when max degrees of numerator and denominator are 3), composing ( f(f(x)) ) results in a rational function whose overall degree is bounded—making ( f(f(x)) - x ) a rational equation with a numerator polynomial of degree at most 9. This structure is foundational in analyzing fixed points and functional behavior in higher-degree rational systems.", "---", "Keywords: rational function, degree of rational function, composition of rational functions, degree bounded by max num/den, ( f(f(x)) ), polynomial degree 9, rational equation, analytic algebra, functional degree bounds."]