Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x}{x^{2}-x - 12} ).
determine the oblique asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one oblique asymptote,
(type an equation. use integers or fractions for any numbers in the equation.)
b. the function has two oblique asymptotes. the oblique asymptote with a negative slope is, and the oblique asymptote with a positive slope is
(type equations. use integers or fractions for any numbers in the equations.)
c. there is no oblique asymptote.
determine the points, if any, at which the graph of ( r ) intersects the horizontal or oblique asymptote, if one exists. select the correct choice 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. use a comma to separate answers as needed.)
b. the graph of ( r ) intersects the horizontal or oblique asymptote at infinitely many points.
c. there is no point at which the graph of ( r ) intersects the horizontal or oblique asymptote.
d. there is no horizontal or oblique asymptote.

Explanation:

Step1: Determine the degree of numerator and denominator

The degree of the numerator \(n = 1\) (for \(y=x\)), and the degree of the denominator \(m=2\) (for \(y = x^{2}-x - 12\)).

Step2: Use the rule for oblique asymptotes

For a rational function \(R(x)=\frac{f(x)}{g(x)}\), if \(n

Step3: Find the horizontal asymptote

For a rational function \(R(x)=\frac{f(x)}{g(x)}\) with \(n < m\), the horizontal asymptote is \(y = 0\).
Set \(\frac{x}{x^{2}-x - 12}=0\). Solving \(x=0\) (since \(x^{2}-x - 12=(x - 4)(x+3)
eq0\) when \(x = 0\)).

Answer:

C. There is no oblique asymptote.
A. The graph of \(R\) intersects the horizontal or oblique asymptote at \((0,0)\)