Rational Functions

Exploring the Nature of Rational Functions

Rational functions are defined as the quotient of two polynomial functions, represented by the formula \( y = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. A polynomial is an expression comprising variables with non-negative integer exponents and constant coefficients. For instance, \( y = x^5 + 6x^3 + 2x + 9 \) exemplifies a polynomial. In rational functions, the denominator \( Q(x) \) must never equal zero, as division by zero is undefined in mathematics. This condition introduces the concept of a function's domain, which excludes values that would make the denominator zero, leading to discontinuities or undefined points in the function.

Graphical Characteristics of Rational Functions and Asymptotes

Graphing rational functions reveals their behavior through the presence of asymptotes—lines that the graph approaches but does not intersect. Asymptotes can be vertical, horizontal, or slant (oblique). Vertical asymptotes occur at values of \( x \) that make the denominator \( Q(x) \) zero, indicating where the function's value becomes unbounded. Horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the numerator's degree is less, the horizontal asymptote is \( y = 0 \). If the numerator's degree is higher, the function does not have a horizontal asymptote and may exhibit end behavior similar to that of a polynomial of the corresponding degree.