Functions - Properties of Rational Functions

Definition and Domain: A rational function is defined as $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials. The domain consists of all real numbers except those where $Q(x) = 0$. Visually, these exclusions create breaks or gaps in the graph, representing values where the function is undefined.

Vertical Asymptotes (VA): These occur at values of $x$ where the denominator $Q(x) = 0$ but the numerator $P(x) \neq 0$ (after simplification). On a graph, the curve will get closer and closer to these vertical dashed lines, moving toward $+\infty$ or $-\infty$, but will never touch or cross them.

Horizontal Asymptotes (HA): These indicate the end behavior of the function as $x \to \pm\infty$. If the degree of the numerator $n$ is less than the degree of the denominator $m$, the HA is $y = 0$. If $n = m$, the HA is the ratio of the leading coefficients. Visually, the graph approaches this horizontal line at the far left and right edges.

Oblique (Slant) Asymptotes: When the degree of the numerator is exactly one higher than the degree of the denominator, the graph follows a linear path $y = ax + b$ as $x$ becomes very large or very small. This line is found using polynomial long division. The graph will approach this diagonal line instead of a horizontal one.

Holes (Removable Discontinuities): A hole occurs at $x = c$ if $(x - c)$ is a common factor in both the numerator and denominator. Graphically, the curve appears continuous, but there is a single point missing, represented by an open circle $\circ$ at that specific coordinate.

Intercepts: The $y$-intercept is found by evaluating $f(0)$, showing where the graph crosses the vertical axis. The $x$-intercepts (zeros) are the roots of the numerator $P(x) = 0$ (provided they are not also roots of the denominator), showing where the graph crosses the horizontal axis.

Sign Diagrams and Behavior: To understand the shape of the graph, we use critical values (zeros and VAs) to test intervals. This tells us if the graph is above ($f(x) > 0$) or below ($f(x) < 0$) the $x$-axis, helping to determine how the curve approaches its asymptotes.

Transformations of $\frac{1}{x}$: Rational functions in the form $f(x) = \frac{a}{x-h} + k$ are shifts of the parent function $y = \frac{1}{x}$. The value $h$ represents a horizontal shift (and the VA), $k$ represents a vertical shift (and the HA), and $a$ represents a vertical stretch or reflection.