Example Question - function behavior
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Analysis of Rational Function for Asymptotes and Domain
<p>To determine the domain of f(x), find the x-values for which the denominator is non-zero:</p> <p>\[ x^2 - 1 \neq 0 \]</p> <p>\[ (x + 1)(x - 1) \neq 0 \]</p> <p>\[ x \neq \pm1 \]</p> <p>Thus, the domain of f(x) is all real numbers except x = -1 and x = 1.</p> <p>Next, identify any vertical asymptotes by setting the denominator equal to zero:</p> <p>\[ x^2 - 1 = 0 \]</p> <p>\[ x = \pm1 \]</p> <p>Therefore, there are vertical asymptotes at x = -1 and x = 1.</p> <p>To find horizontal asymptotes, examine the degrees of the numerator and denominator. Since the degree of the numerator (2) is the same as the degree of the denominator (2), compute the ratio of the leading coefficients:</p> <p>\[ \lim_{{x \to \pm\infty}} \frac{3x^2}{x^2-1} = 3 \]</p> <p>There is a horizontal asymptote at y = 3.</p> <p>Lastly, analyze the behavior of the function near the asymptotes. As x approaches 1 or -1 from the left or right, the function f(x) diverges to positive or negative infinity, depending on the direction of approach. The function approaches the horizontal asymptote y = 3 as x approaches plus or minus infinity.</p>
Analysis of Function Domain, Asymptotes, and Behavior
<p>The domain of \( f(x) \) is all real numbers except where the denominator equals zero. Set the denominator equal to zero and solve for x:</p> <p>\( x^2 - 1 = 0 \)</p> <p>\( (x + 1)(x - 1) = 0 \)</p> <p>\( x = 1 \) or \( x = -1 \)</p> <p>So the domain is \( x \in \mathbb{R} \), \( x \neq 1 \), \( x \neq -1 \).</p> <p>To find vertical asymptotes, look at the points where the function is undefined, which are at \( x = 1 \) and \( x = -1 \).</p> <p>To find the horizontal asymptote, examine the degrees of the numerator and denominator:</p> <p>The degree of the numerator (2) is equal to the degree of the denominator (2). The horizontal asymptote is the ratio of the leading coefficients:</p> <p>\( y = \frac{2}{1} = 2 \)</p> <p>Analyze the behavior around the vertical asymptotes:</p> <p>As \( x \) approaches \( 1 \) from the left, \( f(x) \) goes to \( -\infty \).</p> <p>As \( x \) approaches \( 1 \) from the right, \( f(x) \) goes to \( +\infty \).</p> <p>As \( x \) approaches \( -1 \) from the left, \( f(x) \) goes to \( +\infty \).</p> <p>As \( x \) approaches \( -1 \) from the right, \( f(x) \) goes to \( -\infty \).</p> <p>The function approaches the horizontal asymptote \( y = 2 \) as \( x \) goes to \( \pm\infty \).</p>
Analysis of Function Behavior Based on its Graph
Unfortunately, the image provided does not display the graph of the function \( f(x) \). To answer the questions posed (A and B), we need the graph to determine where \( f(x) \) is positive and whether any of the statements about the values of \( f(x) \) are true. Without the graph, we cannot accurately solve questions A and B. Please provide the graph or more information so that I can assist you further.
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