Rational function

A rational function is a function expressible as the quotient of two polynomials, with a denominator that is not the zero polynomial. Its domain excludes inputs where the denominator evaluates to zero.

Polynomial: An expression formed from constants and variables using addition, subtraction, and nonnegative integer powers. The numerator and denominator are polynomials.
Vertical asymptote: A vertical line the graph of a function approaches without bound near some input. An uncanceled denominator zero often produces this behavior.
Inverse proportion: A relationship in which one quantity varies as the reciprocal of another. The simplest reciprocal model is a rational function.
Polynomial function: A function defined by a polynomial in its input. A polynomial has no denominator-induced domain restrictions or poles.
Domain of a function: The set of inputs for which a function is defined. Denominator zeros determine which inputs must be excluded.
Horizontal asymptote: A horizontal line that a function's graph approaches as its input tends to positive or negative infinity. Comparing polynomial degrees often determines the limiting height.
Rational equation: An equation in which one or more expressions are rational functions of the unknown. Solving often requires clearing denominators and checking excluded inputs.
Algebraic function: A function satisfying a polynomial equation whose coefficients are polynomials in its input. Every rational function is algebraic, but algebraic functions need not be rational.
Factorization: The expression of a mathematical object as a product of simpler factors. Factoring numerator and denominator reveals cancellations and excluded values.
Oblique asymptote: A slanted line that a function's graph approaches as its input tends to infinity. Polynomial division can reveal a linear asymptote when the numerator degree exceeds the denominator degree by one.

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