Question: Follow the steps for graphing a rational function to graph the function \( R(x) = \frac{x^2 + 5x…

Follow the steps for graphing a rational function to graph the function \( R(x) = \frac{x^2 + 5x - 36}{x^2 - x - 12} \).

Determine the behavior of the graph on either side of any vertical asymptotes, if one exists. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

A. It approaches \( \infty \) on one side of the asymptote(s) at \( x = \) and \( -\infty \) on the other. It approaches either \( \infty \) or \( -\infty \) on both sides of the asymptote(s) at \( x = \) (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.)

B. It approaches \( \infty \) on one side of the asymptote(s) at \( x = \) and \( -\infty \) on the other. (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.)

C. It approaches either \( \infty \) or \( -\infty \) on both sides of the asymptote(s) at \( x = \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.)

D. The function has no vertical asymptote.

Solution

The problem involves determining the behavior of the graph of a rational function on either side of its vertical asymptotes. The given function is \[ R(x) = \frac{x^2 + 5x - 36}{x^2 - x - 12} \] First, find the vertical asymptotes by setting the denominator equal to zero. \[ x^2 - x - 12 = 0 \] Factor the quadratic equation. \[ x^2 - x - 12 = (x - 4)(x + 3) = 0 \] Set each factor equal to zero. \[ \begin{cases} x - 4 = 0 & \\ x + 3 = 0 & \end{cases} \] Solve for \(x\). \[ \begin{cases} x = 4 & \\ x = -3 & \end{cases} \] So, the vertical asymptotes are at \(x = 4\) and \(x = -3\). Now, consider the behavior of the function as it approaches these asymptotes. We look at the limits from either side of each asymptote. For \(x = 4\): Consider \(x \to 4^-\) and \(x \to 4^+\). For \(x = -3\): Consider \(x \to -3^-\) and \(x \to -3^+\). Since the asymptotes are vertical, the function will approach either \(+\infty\) or \(-\infty\) on either side, depending on the signs of the numerator and denominator. Therefore, the answer is B. It approaches \(+\infty\) on one side of the asymptote(s) at \(x = 4, -3\) and \(-\infty\) on the other.