Question

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

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 \(u = x + 2\), \(u^\prime=1\), \(v=x - 2\), \(v^\prime = 1\).

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

Since \((x - 2)^{2}>0\) for all \(x
eq2\), and \(f^\prime(x)=\frac{-4}{(x - 2)^{2}}<0\) for all \(x
eq2\)

Answer:

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