sketch the graph of the following function. indicate where the function is increasing or decreasing, where…

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.\n f(x)=\frac{x + 1}{x^{2}-6x - 9}\n(simplify your answer. type your answer in interval - notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nc. the function is increasing on (square). the function is never decreasing.\n(simplify your answer. type your answer in interval - notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nd. the function is never increasing or decreasing.\ndetermine the coordinates of the relative extrema. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the coordinates of the relative extrema are (square).\n(type an ordered pair. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. there are no relative extrema.\ndetermine the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function has two vertical asymptotes. the leftmost asymptote is (square) and the rightmost asymptote is (square).\n(type equations.)\nb. the function has one vertical asymptote (square). (type an equation.)\nc. the function has no vertical asymptotes.

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.\n f(x)=\frac{x + 1}{x^{2}-6x - 9}\n(simplify your answer. type your answer in interval - notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nc. the function is increasing on (square). the function is never decreasing.\n(simplify your answer. type your answer in interval - notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nd. the function is never increasing or decreasing.\ndetermine the coordinates of the relative extrema. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the coordinates of the relative extrema are (square).\n(type an ordered pair. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. there are no relative extrema.\ndetermine the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function has two vertical asymptotes. the leftmost asymptote is (square) and the rightmost asymptote is (square).\n(type equations.)\nb. the function has one vertical asymptote (square). (type an equation.)\nc. the function has no vertical asymptotes.

Answer

Explanation:

Step1: Simplify the denominator

First, factor the denominator $x^{2}-8x - 9=(x - 9)(x+1)$. So the function is $f(x)=\frac{x + 1}{(x - 9)(x + 1)}$. After canceling out the common - factor $(x + 1)$ (for $x\neq - 1$), we get $f(x)=\frac{1}{x - 9},x\neq - 1$.

Step2: Find the domain

The domain of the function is all real numbers except $x = 9$ and $x=-1$.

Step3: Find intervals of increase and decrease

Differentiate $y=\frac{1}{x - 9}=(x - 9)^{-1}$ using the power - rule. The derivative $y^\prime=-(x - 9)^{-2}=-\frac{1}{(x - 9)^{2}}$. Since $y^\prime<0$ for all $x$ in the domain of the function (except $x = 9$ and $x=-1$), the function is decreasing on $(-\infty,-1)\cup(-1,9)\cup(9,\infty)$ and never increasing.

Step4: Find relative extrema

Since the derivative $y^\prime=-\frac{1}{(x - 9)^{2}}\neq0$ for all $x$ in the domain of the function, there are no relative extrema.

Step5: Find vertical asymptotes

Set the denominator of the simplified function equal to zero. For $y=\frac{1}{x - 9}$, when $x = 9$, the function has a vertical asymptote. The equation of the vertical asymptote is $x = 9$.

Answer:

  1. The function is decreasing on $(-\infty,-1)\cup(-1,9)\cup(9,\infty)$; the function is never increasing.
  2. There are no relative extrema.
  3. The function has one vertical asymptote $x = 9$.