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Question

Feature
A function is graphed on the coordinate plane.The graph consists of a curve with an asymptote at x equals negative 2. The function decreases for x-values on the interval left parenthesis negative 2 comma 0 right parenthesis and increases for x-values on the interval left parenthesis 0 comma infinity right parenthesis.
Question
Use the graph to answer the question.
Which statement describes why x equals negative 2 is an asymptote for the graphed function?
(1 point)
Responses

f left-parenthesis x right-parenthesis right arrow negative 2 as x right arrow infinity
Image with alt text: f left-parenthesis x right-parenthesis right arrow negative 2 as x right arrow infinity

f left-parenthesis x right-parenthesis right arrow infinity as x right arrow negative 2
Image with alt text: f left-parenthesis x right-parenthesis right arrow infinity as x right arrow negative 2

f left-parenthesis x right-parenthesis right arrow negative infinity as x right arrow negative 2
Image with alt text: f left-parenthesis x right-parenthesis right arrow negative infinity as x right arrow negative 2

f left-parenthesis x right-parenthesis right arrow negative 2 as x right arrow negative infinity

Answer 1

To determine why \( x = -2 \) is an asymptote for the graphed function, we need to analyze the behavior of the function as \( x \) approaches \(-2\).

Given that the function decreases for \( x \) values on the interval \((-2, 0)\) and increases for \( x \) values on the interval \((0, \infty)\), this suggests that the function approaches a specific value as \( x \) approaches -2 from the right (i.e., as \( x \) approaches \(-2\) from values greater than \(-2\)), and diverges as \( x \) approaches -2 from the left (values lesser than -2).

In this case, the correct statement describing the behavior at the asymptote is:

"f(x) approaches infinity as x approaches negative 2."

So the appropriate response is:

f(x) → infinity as x → -2.