Verified Answer :
- Identify the vertical asymptotes at \(x = 1\) and \(x = -1\) and the horizontal asymptote at \(y = 0\).
- Determine the intercept at the origin \((0,0)\).
- Recognize that the function is odd (symmetric about the origin).
- Conclude that the graph is defined by three branches: a curve in the upper right approaching the axes, a central curve passing through the origin, and a curve in the lower left approaching the axes. \(\boxed{\text{Three-branched curve with asymptotes at } x=\pm 1 \text{ and } y=0}\).
Explanation
- Problem Analysis and Key Features
We analyze the function \(f(x) = \frac{2x}{x^2 - 1}\) to determine its graph.
- Vertical Asymptotes: The function is undefined when the denominator is zero, \(x^2 - 1 = 0\), which occurs at \(x = 1\) and \(x = -1\). These are the vertical asymptotes of the graph.
- Horizontal Asymptote: As \(x\) approaches positive or negative infinity, the degree of the denominator is greater than the degree of the numerator. Thus, the limit \(\lim_{x \to \pm \infty} \frac{2x}{x^2 - 1} = 0\). The horizontal asymptote is the line \(y = 0\) (the x-axis).
- Intercepts: The function equals zero when the numerator is zero, \(2x = 0\), giving \(x = 0\). Thus, the graph passes through the origin at \((0, 0)\), which is both the x-intercept and the y-intercept.
- Symmetry: By replacing \(x\) with \(-x\), we check for symmetry: \(f(-x) = \frac{2(-x)}{(-x)^2 - 1} = \frac{-2x}{x^2 - 1} = -f(x)\). Since \(f(-x) = -f(x)\), the function is odd, meaning the graph has rotational symmetry about the origin.
- Behavior and Interval Analysis
To understand the behavior of the function across its domain, we examine the intervals defined by the vertical asymptotes:
- Interval \((1, \infty)\): In this region, \(x^2 > 1\), so \(f(x) > 0\). As \(x\) increases towards infinity, \(f(x)\) approaches \(0\) from above. As \(x\) approaches \(1\) from the right, the denominator is positive and near zero, so \(f(x)\) approaches \(+\infty\).
- Interval \((-1, 1)\): Here, the function passes through the origin \((0,0)\). As \(x\) approaches \(1\) from the left, \(f(x)\) approaches \(-\infty\). As \(x\) approaches \(-1\) from the right, \(f(x)\) approaches \(+\infty\).
- Interval \((-\infty, -1)\): In this region, \(x^2 > 1\) but \(x < 0\), so \(f(x) < 0\). As \(x\) approaches negative infinity, \(f(x)\) approaches \(0\) from below. As \(x\) approaches \(-1\) from the left, \(f(x)\) approaches \(-\infty\).
Combining these features, the graph consists of three distinct branches consistent with the identified asymptotes, intercepts, and symmetry.
- Summary of Graph Characteristics
The graph of \(f(x) = \frac{2x}{x^2 - 1}\) is characterized by vertical asymptotes at \(x = -1\) and \(x = 1\), a horizontal asymptote at \(y = 0\), and an intercept at \((0,0)\). The function is positive for \(x > 1\) and \(-1 < x < 0\), and negative for \(x < -1\) and \(0 < x < 1\). The symmetry about the origin confirms the shape of the graph, which exhibits the characteristic "hyperbolic" branches split by the vertical asymptotes. The final answer describing the graph is defined by these properties.
Examples
Understanding rational functions like \(f(x) = \frac{2x}{x^2 - 1}\) is essential in fields like electrical engineering, where such functions often describe the impedance or transfer function of a circuit. By analyzing asymptotes and symmetry, engineers can predict the circuit's behavior at extreme frequencies (where the function nears its horizontal asymptote) or near resonance points (where the function approaches vertical asymptotes), helping them design more stable and efficient systems.