Question

sketch the graph of the following function. indicate where the function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
f(x)=\frac{x + 1}{x^{2}-8x - 9}
the function has one horizontal asymptote, y = 0 (type an - equation)
a. the function has two vertical asymptotes. the left - most asymptote is
and the right - most asymptote is
(type equations.)
b. the function has one vertical asymptote, x = 9 (type an equation)
c. the function has no vertical asymptotes.
determine the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the function has one horizontal asymptote, y = 0 (type an equation)
b. the function has two horizontal asymptotes. the top asymptote is
and the bottom asymptote is
(type equations.)
c. the function has no horizontal asymptotes.
determine the slant asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the slant asymptote(s) is(are)
(type an equation. use a comma to separate answers as needed.)
b. the function has no slant asymptotes.

Explanation:

Step1: Find vertical asymptotes

Set the denominator equal to zero. Given the function \(f(x)=\frac{x + 1}{x^{2}-9}\), we solve \(x^{2}-9=0\). Factoring, we get \((x + 3)(x - 3)=0\). So \(x=-3\) and \(x = 3\) are the vertical asymptotes.

Step2: Find horizontal asymptotes

Since the degree of the numerator (\(n = 1\)) is less than the degree of the denominator (\(m=2\)), the horizontal - asymptote is \(y = 0\).

Step3: Find slant asymptotes

Since the degree of the numerator is less than the degree of the denominator, there are no slant asymptotes.

Answer:

For vertical asymptotes: The function has two vertical asymptotes. The left - most asymptote is \(x=-3\) and the right - most asymptote is \(x = 3\).
For horizontal asymptotes: The function has one horizontal asymptote, \(y = 0\).
For slant asymptotes: The function has no slant asymptotes.