A function is graphed on the coordinate plane.The function graphed has 3 branches. The first branch starts just above the x-axis and passes through the approximate points left parenthesis negative 3 comma 0.25 right parenthesis, left parenthesis negative 2 comma 0.667 right parenthesis, and left parenthesis negative 1.5 comma 1.6 right parenthesis, approaching but never reaching x equals negative 1. The second branch starts at the bottom of the graph just to the right of x equals negative 1 and passes through the approximate points left parenthesis negative 0.75 comma negative 4.571 right parenthesis, left parenthesis negative 0.5 comma negative 2.667 right parenthesis, left-parenthesis 0 comma negative 2 right-parenthesis, left parenthesis 0.5 comma negative 2.667 right parenthesis, and left parenthesis 0.75 comma negative 4.571 right parenthesis approaching but never reaching x equals 1. The third branch starts just to the right of x equals 1 and passes through the approximate points left parenthesis 1.5 comma 1.6 right parenthesis, left parenthesis 2 comma 0.667 right parenthesis, and left parenthesis 3 comma 0.25 right parenthesis approaching but never reaching the x-axis.

Question

Question
Use the graph to answer the question.
Find the interval(s) over which the function is increasing.
(1 point)
Responses

left parenthesis 0 comma 1 right parenthesis union left parenthesis 1 comma infinity right parenthesis
Image with alt text: left parenthesis 0 comma 1 right parenthesis union left parenthesis 1 comma infinity right parenthesis

left parenthesis negative infinity comma negative 1 right parenthesis union left parenthesis negative 1 comma 0 right parenthesis
Image with alt text: left parenthesis negative infinity comma negative 1 right parenthesis union left parenthesis negative 1 comma 0 right parenthesis

left parenthesis negative infinity comma negative 1 right parenthesis
Image with alt text: left parenthesis negative infinity comma negative 1 right parenthesis

left parenthesis 1 comma infinity right parenthesis

Answer 1

To determine the intervals over which the function is increasing, we need to analyze the branches of the graph described.

1. First Branch: This branch starts above the x-axis and passes through points as it approaches \( x = -1 \). As \( x \) increases from negative infinity to just before \( x = -1 \), the function is increasing (since the y-values are getting larger).

2. Second Branch: This branch starts just to the right of \( x = -1 \) and decreases as \( x \) moves from just greater than \(-1\) to just before \( x = 1\). Therefore, the function is decreasing in this interval.

3. Third Branch: This branch starts just to the right of \( x = 1 \) and increases as \( x \) goes from just greater than \( 1 \) towards positive infinity, passing through points that show an increase in the y-values.

From this analysis, we can conclude the function is increasing in the following intervals:

• From \( (-\infty, -1) \)
• From \( (1, \infty) \)

Therefore, the correct response is:

left parenthesis negative infinity comma negative 1 right parenthesis union left parenthesis 1 comma infinity right parenthesis