What defines a rational function?

A rational function is defined as a function that can be expressed as the quotient of two polynomial functions, specifically in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) is not equal to zero. This definition is important because if q(x) were to equal zero for any value of x, it would create an undefined expression, which is not permissible in the context of rational functions.

The correct answer emphasizes this crucial aspect of rational functions. In this form, both the numerator and the denominator can be any polynomial, and the function as a whole can exhibit a wide range of behaviors, including asymptotes, intercepts, and continuity in specific intervals. Understanding this definition is foundational for exploring further characteristics of rational functions, such as their graphs and properties.

The other options illustrate different types of functions: a quadratic function, a trigonometric function, and an exponential function, none of which meet the specific criterion of being a rational function involving the quotient of polynomials, thus differentiating them from rational functions.