Question

find the asymptotes and intercepts of the function, and graph the function.
$g(x)=\frac{x - 1}{x^2 - 2x - 24}$

find the horizontal asymptotes. select the correct answer below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one horizontal asymptote, \boxed{}.
(type an equation.)
b. the function has two horizontal asymptotes. the top asymptote is \boxed{} and the bottom asymptote is \boxed{}.
(type equations.)
c. the function has no horizontal asymptotes.
find the vertical asymptotes. select the correct answer below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has two vertical asymptotes. the leftmost asymptote is \boxed{} and the rightmost asymptote is \boxed{}.
(type equations.)
b. the function has one vertical asymptote, \boxed{}.
(type an equation.)
c. the function has no vertical asymptotes.

Explanation:

Step1: Find Horizontal Asymptote

For a rational function \( g(x)=\frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator and \( D(x) \) is the denominator, we compare the degrees of \( N(x) \) and \( D(x) \). The degree of \( N(x)=x - 1 \) is \( 1 \) (since the highest power of \( x \) is \( 1 \)), and the degree of \( D(x)=x^{2}-2x - 24 \) is \( 2 \). When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \). So the function has one horizontal asymptote \( y = 0 \).

Step2: Find Vertical Asymptotes

First, factor the denominator: \( x^{2}-2x - 24=(x - 6)(x + 4) \). Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points). Set \( D(x)=0 \): \( (x - 6)(x + 4)=0 \), so \( x = 6 \) or \( x=-4 \). Check the numerator at these points: \( N(6)=6 - 1 = 5
eq0 \) and \( N(-4)=-4 - 1=-5
eq0 \). So there are two vertical asymptotes, \( x=-4 \) (leftmost) and \( x = 6 \) (rightmost).

Answer:

Horizontal Asymptote:

A. The function has one horizontal asymptote, \( \boldsymbol{y = 0} \).

Vertical Asymptotes:

A. The function has two vertical asymptotes. The leftmost asymptote is \( \boldsymbol{x=-4} \) and the rightmost asymptote is \( \boldsymbol{x = 6} \).