Rational Functions and Asymptotes
What are Rational Functions?
A rational function is a function that can be written as the ratio of two polynomial functions. The general form of a rational function is:
$f(x) = \frac{P(x)}{Q(x)}$
where $P(x)$ and $Q(x)$ are polynomial functions, and $Q(x) \neq 0$. The domain of a rational function includes all real numbers except the values of $x$ that make the denominator equal to zero.
Domain of Rational Functions
To find the domain of a rational function:
- Set the denominator equal to zero: $Q(x) = 0$
- Solve for $x$
- Exclude these values from the domain
The domain is typically written as: $\{x \in \mathbb{R} | Q(x) \neq 0\}$
Understanding Asymptotes
Asymptotes are lines that the graph of a function approaches but never touches or crosses (except in special cases). Rational functions can have three types of asymptotes:
1. Vertical Asymptotes
Occur at values where the denominator equals zero (and the numerator doesn't). The function approaches infinity or negative infinity as $x$ approaches these values.
2. Horizontal Asymptotes
Describe the behavior of the function as $x$ approaches positive or negative infinity. They occur when the degree of the numerator is less than or equal to the degree of the denominator.
3. Slant (Oblique) Asymptotes
Occur when the degree of the numerator is exactly one more than the degree of the denominator. The function approaches a slanted line as $x$ approaches infinity.