A rational function is a quotient of polynomial functions:

$$f(x)=\frac{P(x)}{Q(x)},\qquad Q(x)\ne0.$$

The denominator makes the domain an essential part of the function.

Domain and excluded inputs

Every real value that makes $Q(x)=0$ must be excluded from the domain. For example,

$$f(x)=\frac{x^2-1}{x-1}$$

is undefined at $x=1$.

Although factoring gives

$$\frac{(x-1)(x+1)}{x-1}=x+1,$$

the original function still has no value at $x=1$. Its graph therefore follows the line $y=x+1$ except for a hole at that input.

Zeros and asymptotes

A rational function is zero where its numerator is zero and its denominator is not. Other excluded inputs may produce vertical asymptotes, where the magnitude of the function grows without bound as $x$ approaches the excluded value.

The behavior for large $|x|$ depends on the leading terms of the numerator and denominator. For example,

$$f(x)=\frac1x$$

approaches $0$ as $|x|$ grows, so $y=0$ is a horizontal asymptote.

Algebra and geometry

Factoring and simplifying a rational expression can reveal zeros, holes and asymptotic behavior, but domain restrictions must be carried through every simplification. The graph reflects both the simplified rule and the inputs excluded by the original function.