91Ó°ÊÓ

Graphing rational functions involves plotting points on a Cartesian plane where the function is defined and identifying important features like intercepts and asymptotes. For the function \( f(x) = \frac{1}{x^2 - 1} \), the graph has interesting behaviors based on where \( x \) falls in relation to the domain exclusions and asymptotes.
To effectively graph \( f(x) \):

• Identify critical values, such as zeros of the function (if there are any) and where vertical and horizontal asymptotes occur.
• Evaluate several points in each interval determined by these critical values to understand how the function behaves in each section of the domain.
• Use these points to sketch the graph, showing it approaching but never crossing the asymptotes.

Beginning with the identified asymptotes and plotting key points in each domain interval allows for accurate depiction of the rational function's behavior on a graph.

The domain of a rational function is the set of all possible input values (\( x \) values) for which the function is defined. This generally excludes values that make the denominator zero, as division by zero is undefined. For \( f(x) = \frac{1}{x^2 - 1} \), the denominator \( x^2 - 1 \) is zero when \( x = 1 \) and \( x = -1 \).
Therefore, the domain is all real numbers except \( x = 1 \) and \( x = -1 \). In interval notation, this is \((-\infty, -1) \cup (-1, 1) \cup (1, \infty)\).
The range of a function is the set of all possible output values (\( y \) values). For this function, since the horizontal asymptote is at \( y = 0 \), \( f(x) \) can approach 0 but does not actually take the value of 0. It behaves differently in different parts of the domain:

• For \( x \) in \((-\infty, -1)\) and \((1, \infty)\), \( f(x) > 0 \).
• For \( x \) in \((-1, 1)\), \( f(x) < 0 \).

Consequently, the range is \((-\infty, 0) \cup (0, \infty)\).

Asymptotes are lines that the graph of a function approaches but never actually reaches. They play a crucial role in understanding the behavior of rational functions. There are two primary types of asymptotes: vertical and horizontal.

• Vertical Asymptotes: These occur at the \( x \)-values where the function becomes undefined. For \( f(x) = \frac{1}{x^2 - 1} \), vertical asymptotes are at \( x = 1 \) and \( x = -1 \). The graph moves towards infinity or negative infinity as it nears these values.
• Horizontal Asymptote: This reflects the behavior of \( f(x) \) as \( x \) approaches infinity. Since the degree of the polynomial in the denominator (2) is greater than the degree of the numerator (0), the horizontal asymptote for this function is at \( y = 0 \).

Understanding and identifying these asymptotes are essential steps in sketching the graph accurately and interpreting the function's behavior.

The end behavior of a rational function describes how the function behaves as \( x \) approaches infinity or negative infinity. This is often influenced by any horizontal asymptotes and the degrees of the numerator and denominator polynomials.
For the function \( f(x) = \frac{1}{x^2 - 1} \), the end behavior is dictated by the horizontal asymptote at \( y = 0 \). As \( x \) approaches either positive or negative infinity:

• The function values get closer and closer to zero.
• Despite approaching zero, \( f(x) \) will never actually equal zero. This is part of why the x-axis serves as a horizontal asymptote, shaping the graph's distant behavior.

Understanding the end behavior provides a comprehensive view of the function as it stretches indefinitely in either direction on the graph.