Rational Functions: Properties, Domain, Range, and Asymptotes

Rational Functions

Definition of Rational Functions

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

• Definition: $F(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Example: $F(x) = \frac{4x^2 + 3x + 2}{9x^5}$

Domain of Rational Functions

The domain of a rational function is all real numbers except where the denominator is zero.

• Rule: Exclude values of $x$ where $Q(x) = 0$.

• Example: $F(x) = \frac{x^4}{x}$ has domain $\mathbb{R} \setminus \{0\}$.

• Example: $F(x) = \frac{3x}{x-1}$ has domain $\mathbb{R} \setminus \{1\}$.

• Example: $F(x) = \frac{x^2}{x^2-9}$ has domain $\mathbb{R} \setminus \{-3, 3\}$.

Range of Rational Functions

The range is the set of possible output values. Finding the range often requires analyzing the function's behavior, including asymptotes and intercepts.

• Example: $F(x) = \frac{x^4}{x^9}$ has range $\mathbb{R} \setminus \{0\}$ (since $x^9$ in denominator).

• Example: $F(x) = \frac{x^2}{x}$ has range $\mathbb{R} \setminus \{0\}$.

Zeros, Holes, and Undefined Points

Rational functions can have holes (removable discontinuities) and vertical asymptotes (non-removable discontinuities).

• Hole: Occurs when both numerator and denominator are zero at the same $x$ value, and the factor cancels.

• Vertical Asymptote: Occurs when denominator is zero and the factor does not cancel.

• Example: $F(x) = \frac{x^2-1}{x-1}$ has a hole at $x=1$ (since $x-1$ cancels).

• Example: $F(x) = \frac{x^2}{x^2-9}$ has vertical asymptotes at $x=3$ and $x=-3$.

Basic Properties and Examples

• Cancellation: Do not count cancelled factors when finding domain.

• Example: $F(x) = \frac{x^4}{x} \cdot 4 \rightarrow \frac{x^6}{x}$, domain $\mathbb{R} \setminus \{0\}$.

• Example: $F(x) = \frac{x^2}{x^2-9}$, domain $\mathbb{R} \setminus \{-3, 3\}$.

Asymptotes of Rational Functions

Rational functions can have vertical and horizontal asymptotes.

• Vertical Asymptote: Occurs at $x$ values where denominator is zero and not cancelled by numerator.

• Horizontal Asymptote: Determined by the degrees of numerator ($n$) and denominator ($m$):

  • If $n < m$: Horizontal asymptote at $y=0$.

  • If $n = m$: Horizontal asymptote at $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

  • If $n > m$: No horizontal asymptote (may have an oblique/slant asymptote).

• Example: $F(x) = \frac{9x^5}{x^2+3}$, $n=5$, $m=2$, so no horizontal asymptote.

• Example: $F(x) = \frac{4x^3}{x^6}$, $n=3$, $m=6$, so horizontal asymptote at $y=0$.

• Example: $F(x) = \frac{2x^2+3x+16}{x^2+12}$, $n=m=2$, so horizontal asymptote at $y=\frac{2}{1}=2$.

Intercepts of Rational Functions

• X-intercept: Set numerator equal to zero and solve for $x$.

• Y-intercept: Set $x=0$ and solve for $F(0)$.

• Example: $F(x) = \frac{x^2}{x^2-9}$, x-intercept at $x=0$, y-intercept at $F(0)=0$.

Summary Table: Asymptote Rules

Degree of Numerator ($n$)	Degree of Denominator ($m$)	Horizontal Asymptote
$n < m$	Higher denominator	$y=0$
$n = m$	Equal degrees	$y=\frac{\text{lead coeff of numerator}}{\text{lead coeff of denominator}}$
$n > m$	Higher numerator	None (may be slant/oblique)

Additional info:

• When both numerator and denominator are zero at the same $x$ value, the function has a hole at that point.

• Vertical asymptotes are not affected by cancelled factors.

• Horizontal asymptotes describe end behavior as $x \rightarrow \pm \infty$.