2.7 Asymptotes
1. Definition of Asymptotes
An asymptote describes the directional boundaries that a function progressively approaches but mathematically avoids at extreme bounds.
Vertical Asymptotes (VA): Evaluated at specific domain points $x=a$ where the functional output tends dynamically to $+\infty$ or $-\infty$. In simple fractional structures, this occurs exactly where the denominator equates to zero (providing the numerator is non-zero).
Horizontal Asymptotes (HA): Formed by determining the boundary output as $x$ tends to extremely large positive or negative ranges ($x \rightarrow \pm\infty$). The graph stabilizes approaching the horizontal threshold $y=b$.
2. Rational Functions: $f(x) = \frac{Ax + B}{Cx + D}$
Functions possessing standard linear polynomials in the numerator and denominator reliably yield precisely one vertical and one horizontal asymptote.
- Vertical Asymptote: Derived strictly by solving the denominator $Cx + D = 0 \Rightarrow x = -D/C$.
- Horizontal Asymptote: Derived analytically by evaluating the ratio of leading coefficients: $y = \frac{A}{C}$.
EXAMPLE 1 (Rational Asymptote Derivations)
| Function | Vertical Asymptote | Horizontal Asymptote |
|---|---|---|
| $f(x) = \frac{3x - 7}{x - 5}$ | $x = 5$ | $y = 3$ |
| $f(x) = \frac{3x - 7}{2x - 5}$ | $x = \frac{5}{2}$ | $y = \frac{3}{2}$ |
| $f(x) = \frac{8x - 7}{2x + 4}$ | $x = -2$ | $y = 4$ |
| $f(x) = \frac{7}{x - 5}$ | $x = 5$ | $y = 0$ (coeff ratio $0/1$) |
| $f(x) = \frac{7}{x - 5} + 3$ | $x = 5$ | $y = 3$ (shifts HA up 3) |
EXAMPLE 2 (Inverses)
Let $f(x) = \frac{3x+2}{x-4}$. The inverse evaluates precisely to $f^{-1}(x) = \frac{4x+2}{x-3}$. Notice the strict exchange of domain/range properties:
- For $f(x)$: Domain restricts $x \ne 4$ (VA at $x=4$). HA evaluates to $y=3$ (Range restricts $y \ne 3$).
- For $f^{-1}(x)$: Domain restricts $x \ne 3$ (VA at $x=3$). HA evaluates to $y=4$ (Range restricts $y \ne 4$).
Self-Inverse Notice: A rational function where $f^{-1}(x) = f(x)$ creates symmetric asymptotes, e.g., $f(x) = \frac{2x+3}{x-2}$, where VA is $x=2$ and HA is $y=2$.