Rational functions

Reciprocal function. A function $f$ is called a reciprocal function if

$\displaystyle f(x) = \frac{ k }{x}
$

with constant value $ k \neq 0$. The domain of $f$ is $ D = (-\infty,0)\cup (0,\infty)$, that is, the set of all real values except $ x = 0$.

  1. If $ k > 0$, $f(x)$ increases without bound as $x$ approaches 0 from the right, and $f(x)$ decreases without bound as $x$ approaches 0 from the left. In short, we write

    $\displaystyle f(x) \to \infty$    as $ x \to 0+$    and $\displaystyle \quad
f(x) \to -\infty$    as $ x \to 0-$.

  2. If $ k < 0$, $f(x)$ decrease without bound as $x$ approaches 0 from the right, and $f(x)$ increases without bound as $x$ approaches 0 from the left. In short, we write

    $\displaystyle f(x) \to -\infty$    as $ x \to 0+$    and $\displaystyle \quad
f(x) \to \infty$    as $ x \to 0-$.

1. $ k > 0$   2. $ k < 0$
\includegraphics{lec06a.ps}   \includegraphics{lec06b.ps}

Linear rational functions. A function $f$ is called a linear rational function if

$\displaystyle f(x) = \frac{a x + b}{c x + d}$ (1)

Rational functions. A function $f$ is called a rational function if

$\displaystyle f(x) = \frac{g(x)}{h(x)}$ (2)

with polynomial functions $ g(x)$ of degree $n$ and $ h(x)$ of degree $ k$.



Department of Mathematics
Last modified: 2005-09-29