Rational Functions: Properties, Operations, and Applications

Rational Functions

Definition and Structure

A rational function is any function that can be written as the ratio of two polynomials:

• General Form: $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$.

• Domain: All real numbers except where $q(x) = 0$ (since division by zero is undefined).

Domain of Rational Functions

The domain of a rational function excludes any values of $x$ that make the denominator zero.

• To find the domain, solve $q(x) = 0$ and exclude those values from the set of real numbers.

• Example: For $f(x) = \frac{x}{x^2 - 1}$, the domain is all real numbers except $x = 1$ and $x = -1$.

Reducing Rational Functions

After determining the domain, rational functions can often be simplified by factoring and canceling common factors.

• Cancel factors only if they are not in the domain.

• Example: $f(x) = \frac{x^2 - 4}{x^2 - 4x + 4}$ can be reduced by factoring numerator and denominator.

Holes and Vertical Asymptotes

Rational functions may have holes or vertical asymptotes depending on their factors.

• Hole: Occurs at $x = c$ if both $p(x)$ and $q(x)$ have a common factor $(x - c)$, and $c$ is not in the domain.

• Vertical Asymptote: Occurs at $x = c$ if $q(c) = 0$ and $p(c) \neq 0$.

• Example: $f(x) = \frac{(x-3)(x+2)}{(x-3)(x-1)}$ has a hole at $x = 3$ and a vertical asymptote at $x = 1$.

Operations with Rational Functions

Rational functions can be added, subtracted, multiplied, and divided using standard algebraic rules.

• Add/Subtract: Find a common denominator, combine numerators.

• Multiply: Multiply numerators and denominators directly.

• Divide: Multiply by the reciprocal of the divisor.

• Example: $\frac{2}{x+3} + \frac{1}{x-5}$: Common denominator is $(x+3)(x-5)$.

Long Division of Rational Functions

If the degree of the numerator is greater than or equal to the degree of the denominator, use polynomial long division.

• Process: Divide numerator by denominator to obtain a quotient and a remainder.

• Result: $\frac{p(x)}{q(x)} = s(x) + \frac{r(x)}{q(x)}$ where $s(x)$ is the quotient and $r(x)$ is the remainder.

• Example: $\frac{x^2 + 6x + 5}{x + 2}$

Examples and Applications

• Example 1: $f(x) = \frac{x^2 - 9}{x^2 - 4x + 3}$ Factor numerator: $x^2 - 9 = (x-3)(x+3)$ Factor denominator: $x^2 - 4x + 3 = (x-3)(x-1)$ Hole at $x = 3$, vertical asymptote at $x = 1$.

• Example 2: $f(x) = \frac{x^2 + 2x}{x^2 - 4}$ Domain: $x \neq 2, x \neq -2$ No holes, vertical asymptotes at $x = 2$ and $x = -2$.

Summary Table: Properties of Rational Functions

Property	Description	Example
Domain	All real numbers except where denominator is zero	$f(x) = \frac{1}{x-2}$, domain: $x \neq 2$
Hole	Common factor in numerator and denominator, excluded from domain	$f(x) = \frac{(x-3)(x+2)}{(x-3)(x-1)}$, hole at $x = 3$
Vertical Asymptote	Denominator zero, numerator nonzero	$f(x) = \frac{1}{x-1}$, vertical asymptote at $x = 1$
Operations	Add, subtract, multiply, divide using algebraic rules	$\frac{2}{x+3} + \frac{1}{x-5}$
Long Division	Used when numerator degree ≥ denominator degree	$\frac{x^2 + 6x + 5}{x + 2}$

Additional info: Some examples and explanations were inferred and expanded for clarity and completeness based on standard Precalculus curriculum.