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 points, if any, at which the graph of \( R \) intersects the horizontal or oblique asymptote, if one exists. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The graph of \( R \) intersects the horizontal or oblique asymptote at (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.)

B. There is no point at which the graph of \( R \) intersects the horizontal or oblique asymptote.

C. There is no horizontal or oblique asymptote.

Solution

To solve this, we’ll analyze the rational function: \[ R(x) = \frac{x^2 + 5x - 36}{x^2 - x - 12} \] First, determine if there’s a horizontal or oblique asymptote. The degrees of the numerator and denominator are both 2, so there might be a horizontal asymptote. The horizontal asymptote is found by dividing the leading coefficients of the numerator and denominator: Since: \[ \frac{\text{leading coefficient of } x^2 \text{ in numerator}}{\text{leading coefficient of } x^2 \text{ in denominator}} = \frac{1}{1} = 1 \] The horizontal asymptote is \(y = 1\). Now, determine if the graph intersects the horizontal asymptote \(y = 1\). Set \(R(x) = 1\) and solve for \(x\): \[ \frac{x^2 + 5x - 36}{x^2 - x - 12} = 1 \] Multiply both sides by \(x^2 - x - 12\): \[ x^2 + 5x - 36 = x^2 - x - 12 \] Subtract \(x^2\) from both sides: \[ 5x - 36 = -x - 12 \] Add \(x\) to both sides: \[ 6x - 36 = -12 \] Add 36 to both sides: \[ 6x = 24 \] Divide by 6: \[ x = 4 \] Substitute \(x = 4\) back into \(R(x)\) to confirm it results in 1: \[ R(4) = \frac{4^2 + 5 \times 4 - 36}{4^2 - 4 - 12} = \frac{16 + 20 - 36}{16 - 4 - 12} = \frac{0}{0} \] Here, the calculation indicates an indeterminate form, suggesting \(x = 4\) is a removable discontinuity rather than an intersection. Therefore, the graph \(R\) does not intersect the horizontal asymptote. Answer: B. There is no point at which the graph of \(R\) intersects the horizontal or oblique asymptote.