summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of (…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{x + 2}{x - 2} ).\nfind any vertical asymptotes of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function has one vertical asymptote, ( x = 2 ).\n(type an equation.)\nb. the function has two vertical asymptotes. the leftmost asymptote is ( ) and the rightmost asymptote is ( ).\n(type equations.)\nc. there are no vertical asymptotes.\nfind the intervals where ( f(x) ) is increasing or decreasing. select the correct choice below and fill in the answer box(es) to complete your choice.\n(type your answer in interval notation. use a comma to separate answers as needed.)\na. the function is increasing on ( ). it is never decreasing.\nb. the function is increasing on ( ). it is decreasing on ( ).\nc. the function is decreasing on ( ). it is never increasing.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{x + 2}{x - 2} ).\nfind any vertical asymptotes of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function has one vertical asymptote, ( x = 2 ).\n(type an equation.)\nb. the function has two vertical asymptotes. the leftmost asymptote is ( ) and the rightmost asymptote is ( ).\n(type equations.)\nc. there are no vertical asymptotes.\nfind the intervals where ( f(x) ) is increasing or decreasing. select the correct choice below and fill in the answer box(es) to complete your choice.\n(type your answer in interval notation. use a comma to separate answers as needed.)\na. the function is increasing on ( ). it is never decreasing.\nb. the function is increasing on ( ). it is decreasing on ( ).\nc. the function is decreasing on ( ). it is never increasing.

Answer

Explanation:

Step1: Find the derivative of (f(x))

Use the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}), where (u = x + 2), (u^\prime=1), (v=x - 2), (v^\prime = 1). [ \begin{align*} f^\prime(x)&=\frac{(x + 2)^\prime(x - 2)-(x + 2)(x - 2)^\prime}{(x - 2)^{2}}\ &=\frac{1\times(x - 2)-(x + 2)\times1}{(x - 2)^{2}}\ &=\frac{x-2-x - 2}{(x - 2)^{2}}\ &=\frac{-4}{(x - 2)^{2}} \end{align*} ]

Step2: Analyze the sign of (f^\prime(x))

Since ((x - 2)^{2}>0) for all (x\neq2), and (f^\prime(x)=\frac{-4}{(x - 2)^{2}}<0) for all (x\neq2)

Answer:

C. The function is decreasing on ((-\infty,2)\cup(2,\infty)). It is never increasing.