91Ó°ÊÓ

Understanding the asymptotic behavior of a rational function can be quite interesting. An asymptote represents a line that the graph of the function approaches but never actually touches or crosses. In the world of rational functions, we encounter vertical and horizontal asymptotes most frequently.

In our exercise, the function given is
\[f(x)=\frac{3 x^{2}}{x^{2}-1}\] and we identified that the denominator equals zero for values
\[x = 1\] and
\[x = -1\]. These values indicate the vertical asymptotes of the function, as the function's value grows indefinitely large when approaching these points. It's essential to understand that as we get infinitesimally close to these x-values, the function's value, or 'height,' spikes towards positive or negative infinity, hence defining our vertical asymptotes at \(x = 1\) and \(x = -1\).

This behavior near the asymptotes is crucial for sketching accurate graphs of rational functions and helps us predict the function's behavior in calculus for limits and continuity.

To solve rational equations, like finding the values that make the denominator zero in our exercise, it’s necessary to perform algebraic manipulations. The key steps usually involve finding a common denominator, if dealing with multiple fractions, and ensuring that the equation is simplified to a basic form where the variable can be solved for.

For example, we solved the equation
\[x^{2}-1 = 0\] by recognizing it as a difference of squares that factor into
\[ (x-1)(x+1) = 0\]. This gives us the critical x-values where the function’s denominator would be zero, leading to undefined points in the function. Understanding how to solve these equations allows students to accurately find the domain of rational functions and predict their behavior around critical points.

Interval notation is a succinct way of writing sets of numbers, typically used to denote domains and ranges of functions. The intervals are written with parentheses \( ( ) \) or brackets \[ [ ] \], where parentheses indicate that the endpoint is not included (open interval), and brackets indicate the endpoint is included (closed interval).

In our exercise, the domain of \( f(x) = \frac{3x^{2}}{x^{2}-1} \) in interval notation, excluding the vertical asymptotes at \(x = 1\) and \(x = -1\), is
\[(-\infty, -1) \cup (-1, 1) \cup (1, +\infty)\]. Each interval represents all the x-values where the function exists and is clearly delineated to avoid including the asymptotes. Students would benefit from practicing this notation as it is commonly used in higher mathematics to concisely present solution sets and function behavior.