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}-8x - 9} \non what interval(s) is ( f ) increasing and on what interval(s) is ( f ) decreasing? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function is increasing on (square) and decreasing on (square)\n(simplify your answers. type your answers in interval - notation. type exact answers, using radicals as needed. use a comma to separate answers as needed.)\nb. the function is decreasing on (square). the function is never increasing.\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.

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}-8x - 9} \non what interval(s) is ( f ) increasing and on what interval(s) is ( f ) decreasing? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function is increasing on (square) and decreasing on (square)\n(simplify your answers. type your answers in interval - notation. type exact answers, using radicals as needed. use a comma to separate answers as needed.)\nb. the function is decreasing on (square). the function is never increasing.\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.

Answer

Explanation:

Step1: Factor the denominator

First, factor $x^{2}-8x - 9=(x - 9)(x+1)$. So, $f(x)=\frac{x + 1}{(x - 9)(x + 1)}$. The domain of the function is all real - numbers except $x=9$ and $x=-1$ (after canceling out non - zero common factors). After canceling out $x + 1$ (for $x\neq - 1$), $f(x)=\frac{1}{x - 9},x\neq - 1$.

Step2: Find the derivative

Using the quotient rule, if $y=\frac{u}{v}$ where $u = 1$ and $v=x - 9$, then $y^\prime=\frac{u^\prime v-uv^\prime}{v^{2}}$. Here, $u^\prime = 0$ and $v^\prime=1$. So, $y^\prime=-\frac{1}{(x - 9)^{2}}$.

Step3: Determine increasing and decreasing intervals

Since $y^\prime=-\frac{1}{(x - 9)^{2}}<0$ for all $x$ in the domain of the function (i.e., $x\neq9$ and $x\neq - 1$), the function is decreasing on $(-\infty,-1)\cup(-1,9)\cup(9,\infty)$.

Answer:

B. The function is decreasing on $(-\infty,-1)\cup(-1,9)\cup(9,\infty)$. The function is never increasing.