Question

follow the steps for graphing a rational function to graph the function ( f(x)=\frac{x^{2}-15 x - 16}{x + 4} ).
determine the behavior of the graph on either side of any vertical asymptotes, if any exist. select the correct choice 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=square ) and ( -infty ) on the other. it approaches either ( infty ) or
( -infty ) on both sides of the asymptote(s) at ( x=square ).
(type integers or simplified fractions. use a comma to separate answers as needed. type each answer
only once.)
b. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=square ).
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer
only once.)
c. it approaches ( infty ) on one side of the asymptote(s) at ( x=square ) 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.)
d. there is no vertical asymptote.
determine the horizontal asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es)
to complete your choice.

Explanation:

Step1: Simplify the rational function

First, factor the numerator \(x^{2}-15x - 16=(x + 1)(x-16)\). So \(F(x)=\frac{(x + 1)(x - 16)}{x + 4}\). The domain of \(F(x)\) is all real numbers except \(x=-4\) (since the denominator is zero when \(x=-4\)).

Step2: Analyze the behavior near \(x=-4\)

Let's find the one - sided limits as \(x\to - 4\).
For \(x\to - 4^{-}\) (approaching from the left), let \(x=-4-h\) where \(h>0\) and \(h\to0\).
\(F(x)=\frac{(-4 - h+1)(-4 - h - 16)}{-4 - h+4}=\frac{(-3 - h)(-20 - h)}{-h}=\frac{(3 + h)(20 + h)}{-h}\approx\frac{3\times20}{-h}\to-\infty\) as \(h\to0\).
For \(x\to - 4^{+}\) (approaching from the right), let \(x=-4 + h\) where \(h>0\) and \(h\to0\).
\(F(x)=\frac{(-4 + h+1)(-4 + h - 16)}{-4 + h+4}=\frac{(-3 + h)(-20 + h)}{h}=\frac{(3 - h)(20 - h)}{h}\approx\frac{3\times20}{h}\to\infty\) as \(h\to0\).

Answer:

C. It approaches \(\infty\) on one side of the asymptote(s) at \(x = - 4\) and \(-\infty\) on the other.