E2,E3

Rational Functions and Their Graphs

Introduction to Rational Functions

A rational function is a function of the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$. The simplest rational function is the reciprocal function $f(x) = \frac{1}{x}$.

• Domain: All real numbers except where the denominator is zero.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is nonzero.

• Horizontal Asymptotes: Describe the end behavior as $x \to \infty$ or $x \to -\infty$.

Table of Values and Graphing

To understand the behavior of $f(x) = \frac{1}{x}$, we can construct a table of values:

x	-3	-2	-1	-0.5	0	0.5	1	2	3
f(x)	-0.33	-0.5	-1	-2	undefined	2	1	0.5	0.33

As $x$ approaches 0 from the positive side, $f(x) \to +\infty$. As $x$ approaches 0 from the negative side, $f(x) \to -\infty$.

Asymptotes and End Behavior

• Vertical Asymptote: $x = 0$ for $f(x) = \frac{1}{x}$

• Horizontal Asymptote: $y = 0$ for $f(x) = \frac{1}{x}$

As $x \to \infty$, $f(x) \to 0$. As $x \to -\infty$, $f(x) \to 0$.

Arrow Notation for End Behavior

Notation	Meaning
$x \to a^+$	x approaches a from the right
$x \to a^-$	x approaches a from the left
$x \to \infty$	x approaches positive infinity
$x \to -\infty$	x approaches negative infinity
$f(x) \to \infty$	f(x) increases without bound
$f(x) \to -\infty$	f(x) decreases without bound

Transformations of Rational Functions

Shifting and Reflecting

Transformations of $f(x) = \frac{1}{x}$ include horizontal and vertical shifts, reflections, and stretches/compressions. For example:

• $f(x) = \frac{1}{x-2}$ shifts the graph right by 2 units (vertical asymptote at $x=2$).

• $f(x) = \frac{1}{x} + 3$ shifts the graph up by 3 units (horizontal asymptote at $y=3$).

• $f(x) = -\frac{1}{x}$ reflects the graph across the x-axis.

General Form and Asymptotes

The general form $f(x) = \frac{a}{x-h} + k$ has:

• Vertical Asymptote: $x = h$

• Horizontal Asymptote: $y = k$

• Reflection: If $a < 0$, the graph is reflected across the x-axis.

Examples

• Example 1: $f(x) = \frac{1}{x-2}$ Vertical asymptote: $x=2$ Horizontal asymptote: $y=0$ Domain: $(-\infty, 2) \cup (2, \infty)$ Range: $(-\infty, 0) \cup (0, \infty)$

• Example 2: $f(x) = \frac{1}{x-2} + 3$ Vertical asymptote: $x=2$ Horizontal asymptote: $y=3$ Domain: $(-\infty, 2) \cup (2, \infty)$ Range: $(-\infty, 3) \cup (3, \infty)$

• Example 3: $f(x) = -\frac{1}{x-2} + 3$ Vertical asymptote: $x=2$ Horizontal asymptote: $y=3$ Reflection across x-axis.

Limits of Rational Functions

Two-Sided Limits

The two-sided limit $\lim_{x \to a} f(x)$ describes the value that $f(x)$ approaches as $x$ approaches $a$ from both sides. If $f(x)$ approaches the same value from both sides, the two-sided limit exists.

• For $f(x) = \frac{x-2}{x-2}$, as $x \to 2$, $f(x)$ is undefined, but the limit is 1.

One-Sided Limits

The one-sided limit $\lim_{x \to a^+} f(x)$ (from the right) or $\lim_{x \to a^-} f(x)$ (from the left) describes the value $f(x)$ approaches as $x$ approaches $a$ from one side only.

• For $f(x) = \frac{|x-2|}{x-2}$, as $x \to 2^+$, $f(x) \to 1$; as $x \to 2^-$, $f(x) \to -1$.

Infinite Limits

An infinite limit occurs when $f(x)$ increases or decreases without bound as $x$ approaches a certain value. For example, for $f(x) = \frac{1}{(x-2)^2}$, as $x \to 2$, $f(x) \to \infty$ from both sides.

Summary Table: Asymptotes and Limits

Function	Vertical Asymptote	Horizontal Asymptote	Domain	Range
$\frac{1}{x}$	$x=0$	$y=0$	$(-\infty,0)\cup(0,\infty)$	$(-\infty,0)\cup(0,\infty)$
$\frac{1}{x-2}$	$x=2$	$y=0$	$(-\infty,2)\cup(2,\infty)$	$(-\infty,0)\cup(0,\infty)$
$\frac{1}{x-2}+3$	$x=2$	$y=3$	$(-\infty,2)\cup(2,\infty)$	$(-\infty,3)\cup(3,\infty)$

Practice Problems

1. Given $f(x) = \frac{1}{x-2} + 3$, find the equations of the asymptotes, domain, and range.

2. For $f(x) = \frac{1}{x+2} - 1$, describe the transformations and find the asymptotes.

3. Evaluate the following limits using the graph of $f(x)$:

   • $\lim_{x \to 2} f(x)$

   • $\lim_{x \to 2^+} f(x)$

   • $\lim_{x \to 2^-} f(x)$

Additional info: These notes cover the graphical and algebraic analysis of rational functions, including asymptotes, transformations, and limits, which are essential topics in College Algebra.