Rational Function

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) = y_n aⁿ + y_n-1a^n-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,
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.