A rational function is graphed on the coordinate plane.The graph consists of three curves and has asymptotes at y equals 1, x equals negative 2, and x equals 2. The first curve starts at the left of the graph just above y = 1 and increases forever approaching but never reaching x = negative 2. The second curve starts at the bottom of the graph just to the right of x = negative 2, increases until just below the origin, and then decreases forever approaching but never reaching x = 2. The third curve starts at the top of the graph just to the right of x = 2 and decreases forever approaching but never reaching y = 1. There are arrows at both ends of each curve.

Question

Question
Use the graph to answer the question.
Describe the end behavior of the graphed function.
(1 point)
Responses

f left-parenthesis x right-parenthesis right arrow plus-or-minus infinity as x right arrow plus-or-minus 2
Image with alt text: f left-parenthesis x right-parenthesis right arrow plus-or-minus infinity as x right arrow plus-or-minus 2

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

f left-parenthesis x right-parenthesis right arrow infinity as x right arrow 1 and f left-parenthesis x right-parenthesis right arrow negative infinity as x right arrow negative 1
Image with alt text: f left-parenthesis x right-parenthesis right arrow infinity as x right arrow 1 and f left-parenthesis x right-parenthesis right arrow negative infinity as x right arrow negative 1

f left-parenthesis x right-parenthesis right arrow 1 as x right arrow plus-or-minus infinity

Answer 1

To analyze the end behavior of the graphed rational function described, we need to consider the behavior of the function as \( x \) approaches positive and negative infinity, along with the asymptotes given.

1. As \( x \to -2 \) from the left, the first curve approaches \( y = 1 \) but stays above it and does not reach it, indicating that the function goes to \( 1 \).

2. As \( x \to -2 \) from the right, the second curve approaches \( y = 2 \) and decreases indefinitely, indicating that the function heads towards \( -\infty \).

3. As \( x \to 2 \), the third curve decreases towards \( y = 1 \) but never reaches it as you move from positive values approaching 2.

4. As \( x \to -\infty \), the first curve approaches \( y = 1 \).

5. As \( x \to +\infty \), the last curve approaches \( y = 1 \) while decreasing from above.

Given this information, the correct end behavior description of the function is:

\( f(x) \to 1 \) as \( x \to \pm \infty \)

So, the answer to the question is:

f(x) right arrow 1 as x right arrow plus-or-minus infinity.