A rational function is a function that can be expressed as the ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. These functions are defined for all real numbers except where the denominator is zero.
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The domain of a rational function excludes values that make the denominator zero.
Vertical asymptotes occur at the zeros of the denominator if they do not cancel out with zeros in the numerator.
Horizontal asymptotes depend on the degrees of the numerator and denominator polynomials: if they are equal, it is the ratio of their leading coefficients; if the numerator's degree is less, it is $y=0$; if greater, there is no horizontal asymptote.
If a factor cancels out both in the numerator and denominator, it creates a hole in the graph at that point instead of an asymptote.
Rational functions can have oblique (slant) asymptotes if the degree of the numerator is exactly one more than that of the denominator.
Review Questions
How do you determine vertical asymptotes for a rational function?
What happens to a rational function when its numerator and denominator share a common factor?
Describe how to find horizontal or oblique asymptotes for a given rational function.
Related terms
Polynomial: An algebraic expression composed of variables and coefficients involving only non-negative integer powers of variables.
Asymptote: A line that a graph approaches but never touches or crosses. This can be vertical, horizontal, or oblique.
Hole: \text{A point on} \, f(x)\text{'s graph where } \, P(x)\text{ and }\, Q(x)\,\text{cancel each other out causing an undefined value.}