Rational function

Rational function is a quotient of two polynomial functions. It distinguishes from irrational function which cannot be written as a ratio of two polynomials.

Definition
A rational function is a function of the form


 * $$ f(x) = \frac{s(x)}{t(x)}$$

where s and t are polynomial function in x and t is not the zero polynomial. The domain of f is the set of all points x for which the denominator t(x) is not zero.

On the graph restricted values of an axis forms a straight line, called asymptote, which cannot be crossed by the function. If zeros of numerator and denominator are equal, then the function is a horizontal line with the hole on a restricted value of x.

Examples
Let's see an example of $$f(x) = \frac{x^2-x-6x}{x^2+x-20}$$ in a factored form: $$f(x) = \frac{(x+2)(x-3)}{(x+5)(x-4)}$$. Obviously, roots of denominator is -5 and 4. That is, if x takes one of these two values, the denominator becomes equal to zero. Since the division by zero is impossible, the function is not defined or discontinuous at x = -5 and x = 4.

The function is continuous at all other values for x. The domain (area of acceptable values) for the function, as expressed in interval notation, is: $$ (-\infty; -5) \cup (-5; 4) \cup (4; \infty) $$