What is a rational function?

• A. A function expressed as the sum of two polynomial functions
• B. A function that has no real roots
• C. A function that can be expressed as the ratio of two polynomial functions
• D. A function that is always decreasing

Answer: (C)

A rational function is defined as a function that can be expressed as the ratio of two polynomial functions. This means that it takes the form ( f(x) = \frac{P(x)}{Q(x)} ), where ( P(x) ) and ( Q(x) ) are both polynomial expressions and ( Q(x) ) is not equal to zero. By its nature, a rational function can exhibit a range of behaviors, including having vertical asymptotes where the denominator is zero and horizontal asymptotes that describe its end behavior as the input approaches infinity.

The characteristics of rational functions include potential real roots, but they can also have instances where they do not. Additionally, they do not have to adhere to a monotonic behavior such as always decreasing; instead, they can increase or decrease in specific intervals depending on the polynomials involved. Thus, the essence of a rational function lies in its representation as a ratio of polynomials, which is what makes it distinct from other types of functions.