Rational Function Definition Rational Function Definition In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational numbers. Ratio of two polynomial Functions is defined as a rational Function, where, a Polynomial is a finite length expression, made up of 3 basic mathematical operations( addition, subtraction and multiplication and non-negative exponents) which contains variables and constants(like is a polynomial but is not a polynomial) . A function which evaluates a polynomial is known as polynomial function. One argument function named as g , is said to be polynomial if it satisfies , g(a) = yn an + yn-1an-1 + … + y2a2 + y1a + y0 Example : g(a) = a3 – a The function 'a' is said to be rational function only if it can be written as,
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g(a) = R(a) / S(a) where R and S are polynomial Functions in a and S is not zero polynomial. The Set of all points ‘a’ for which the denominator S(a) is not zero comes under the Domain of g. One can assume that this rational function is written in its lowest degree i.e. R and S have many positive degree. Polynomial functions with S(a) = 1, is said to be rational functions. Functions that can be written in this form are not rational functions. Ex : g(a) = sin(a) , is not a rational function. It is not necessary that ‘a’ need to be variable. Example: The rational function g(a) = is defined at a2 = 6 a = . The rational function g(a) = (a2 + 3) / a2 + 1 can’t be defined for the complex numbers but can be defined for Real Numbers. If value of a is a Square root of -1then it Mean the evaluation leads to division by zero. g(b) = (b2 + 2) / (b2 + 1) = (-1 + 3) /(-1 + 1) = 2/ 0, which is undefined. In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. In this setting given a field F and some indeterminate X, a rational expression is any element of the field of fractions of the polynomial ring F[X]. Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique.
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P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. However since F[X] is a unique factorization domain, there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar to how a fraction of integers can always be written uniquely in lowest terms by canceling out common factors. The field of rational expressions is denoted F(X). This field is said to be generated (as a field) over F by (a transcendental element) X, because F(X) does not contain any proper subfield containing both F and the element X.
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