91Ó°ÊÓ

Question: Determine the vertical asymptotes of the rational function \(f(x) = \frac{x^2 - 4}{x^2 - x - 6}\). Answer: To find the vertical asymptotes of the given rational function, follow the steps outlined in the solution. Step 1: Identify the rational function. Here, \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x) = x^2 - 4\) and \(Q(x) = x^2 - x - 6\). Step 2: Find the zeroes of the denominator function (Q(x)). To do this, set Q(x) equal to zero and solve for x: \(x^2 - x - 6 = 0\). Factoring the quadratic equation, we get \((x - 3)(x + 2) = 0\). The zeroes are \(x = 3\) and \(x = -2\). Step 3: Check if the numerator function (P(x)) is not equal to zero at the zeroes of the denominator function. Plug the zeroes of Q(x) into P(x): For \(x = 3\), \(P(3) = (3)^2 - 4 = 5\), which is not equal to zero. For \(x = -2\), \(P(-2) = (-2)^2 - 4 = 0\), which is equal to zero. In this case, we do not have a vertical asymptote at \(x = -2\). Step 4: Write down the vertical asymptotes. The vertical asymptote is located at the x-value where the denominator function is equal to zero and the numerator function is not equal to zero. In this case, the vertical asymptote is at \(x = 3\).