Question

find the asymptotes and intercepts of the function, and graph the function.\\( g(x) = \frac{x - 1}{x^2 - x - 30} \\)\\( \dots \\)find the horizontal asymptotes. select the correct answer below and, if necessary, fill in the answer box(es) to complete your choice\\( \bigcirc \\) a. the function has two horizontal asymptotes. the top asymptote is \\( \square \\) and the bottom asymptote is \\( \square \\).\\( \quad \\) (type equations )\\( \bigcirc \\) b. the function has one horizontal asymptote, \\( \square \\).\\( \quad \\) (type an equation )\\( \bigcirc \\) 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\\( \bigcirc \\) a. the function has one vertical asymptote, \\( \square \\).

Explanation:

Step1: Analyze 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 \( 1 \), and the degree of \( D(x)=x^{2}-x - 30 \) 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: Analyze Vertical Asymptote

First, factor the denominator: \( x^{2}-x - 30=(x - 6)(x + 5) \). The vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points). Set \( (x - 6)(x + 5)=0 \), so \( x = 6 \) and \( x=-5 \). The numerator \( x - 1 \) is not zero at \( x = 6 \) ( \( 6-1 = 5
eq0 \)) and at \( x=-5 \) ( \( -5 - 1=-6
eq0 \) ). So there are two vertical asymptotes \( x = 6 \) and \( x=-5 \), but the question here first asks about horizontal asymptote, so we focus on that.

Answer:

For horizontal asymptote: B. The function has one horizontal asymptote, \( y = 0 \)
For vertical asymptote (if we continue, but based on the first part): The function has two vertical asymptotes \( x = 6 \) and \( x=-5 \), but the first sub - question about horizontal asymptote answer is B with \( y = 0 \)