RATIONAL FUNCTION Meaning and
Definition
-
A rational function, in the context of mathematics, refers to a function that can be expressed as the quotient of two polynomial functions, where the denominator is not equal to zero. It represents a relationship between two variables, where the value of one variable (the dependent variable) depends on the value of the other variable (the independent variable).
A rational function can be written in the general form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials with coefficients (constants) and x represents the independent variable. The numerator, p(x), represents the polynomial function whose degree is typically equal to or less than the degree of the denominator, q(x). The denominator, q(x), is a polynomial with a degree greater than or equal to 1, as the function needs to be non-zero.
The behavior of rational functions can be analyzed by studying their domain, range, asymptotes, and intercepts. The domain consists of all real numbers except the values that make the denominator equal to zero, as these would result in division by zero, which is undefined. The range is the set of all values that the dependent variable can take.
Rational functions often exhibit various characteristics, such as vertical asymptotes (values of x where the function approaches positive or negative infinity), horizontal asymptotes (values of y to which the function approaches as x approaches positive or negative infinity), and any vertical or horizontal intercepts (where the function crosses the x or y-axis, respectively). Understanding these properties helps in graphing rational functions and analyzing their behavior.
Etymology of RATIONAL FUNCTION
The word "rational" in "rational function" comes from the Latin word "rationalis", which means "of reason". This Latin word is derived from the noun "ratio", meaning "reason" or "calculation". In mathematics, a rational function is a function that can be expressed as the quotient or ratio of two polynomials, with the denominator polynomial not being the zero polynomial. The term "rational function" was introduced in the late 19th century, and its usage reflects the idea that these functions involve a ratio or a rational relationship between two polynomial expressions.
