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}
determine the coordinates of the point(s) of inflection. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the coordinates of the point(s) of inflection are
(simplify your answer. type an ordered - pair. use a comma to separate answers as needed.)
b. there are no points of inflection.
determine the x - intercept(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x - intercept(s) is(are)
(type an ordered pair. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)
b. there are no x - intercepts.
determine the y - intercept(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the y - intercept(s) is(are)
(simplify your answer. type an ordered pair. use integers or fractions for any numbers in the expression.)
b. there are no y - intercepts.

Explanation:

Step1: Factor the denominator

First, factor \(x^{2}-8x - 9=(x - 9)(x+1)\). So \(f(x)=\frac{x + 1}{(x - 9)(x + 1)}\), but \(x
eq - 1\). After canceling out the common factor (\(x
eq - 1\)), \(f(x)=\frac{1}{x - 9},x
eq - 1\).

Step2: Find the points of inflection

Find the second - derivative of \(y = f(x)=\frac{1}{x - 9}=(x - 9)^{-1}\).
The first - derivative \(y'=-1\times(x - 9)^{-2}=-\frac{1}{(x - 9)^{2}}\).
The second - derivative \(y'' = 2\times(x - 9)^{-3}=\frac{2}{(x - 9)^{3}}\).
Set \(y'' = 0\), but \(\frac{2}{(x - 9)^{3}}=0\) has no solution. So there are no points of inflection.

Step3: Find the x - intercept

Set \(y = 0\), for \(y=\frac{1}{x - 9}\), \(\frac{1}{x - 9}=0\) has no solution. So there are no x - intercepts.

Step4: Find the y - intercept

Set \(x = 0\), then \(y=\frac{1}{0 - 9}=-\frac{1}{9}\). The y - intercept is \((0,-\frac{1}{9})\).

Answer:

• Points of inflection: There are no points of inflection.
• x - intercept(s): There are no x - intercepts.
• y - intercept(s): \((0,-\frac{1}{9})\)