What are Rational Functions?
A **rational function** is a function that can be written as a ratio (or fraction) of two polynomials. The general form is: $$f(x) = \frac{P(x)}{Q(x)}$$
Where $P(x)$ and $Q(x)$ are both polynomial functions, and importantly, $Q(x)$ is not the zero polynomial.
The key features of rational functions are **asymptotes**, which are lines that the graph approaches but never touches. There are two main types:
- **Vertical Asymptotes**: Occur at values of $x$ where the denominator, $Q(x)$, equals zero, but the numerator, $P(x)$, does not.
- **Horizontal Asymptotes**: Describe the end behavior of the function (what happens as $x$ approaches positive or negative infinity). They depend on the degrees of $P(x)$ and $Q(x)$.
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Video: Asymptotes and Discontinuities!
Watch this video to visualize what rational functions and their asymptotes look like on a graph! 📊
Practice Problems
Find the asymptotes for the following rational functions!
1) What is the vertical asymptote of $f(x) = \frac{1}{x-2}$? (e.g., x=2)
2) What is the horizontal asymptote of $f(x) = \frac{3x+1}{x-5}$? (e.g., y=3)
3) What is the horizontal asymptote of $f(x) = \frac{x^2}{x^3+1}$? (e.g., y=0)
4) What is the vertical asymptote of $f(x) = \frac{x-1}{(x+3)(x-1)}$? (e.g., x=-3)
⚡ Rational Functions Speed Quiz
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