Question

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

To determine the intervals over which the function is increasing, we can analyze the behavior of the branches as described in your question.

1. **First Branch**: This branch starts just above the x-axis and approaches \( x = -1 \) from the left. As x increases from negative infinity towards -1, the function is increasing (since the y-values are increasing).

2. **Second Branch**: This branch starts just below \( x = -1 \) and goes downwards until it reaches \( x = 1 \). Therefore, during this interval, the function is decreasing.

3. **Third Branch**: This branch starts just to the right of \( x = 1 \) and increases as x goes towards positive infinity. Hence, the function is increasing in this interval.

Combining these observations, we find:

- The function is increasing on the interval \( \left( -\infty, -1 \right) \).
- It is also increasing on the interval \( \left( 1, \infty \right) \).

Thus, the overall intervals where the function is increasing are:
\[
\left( -\infty, -1 \right) \cup \left( 1, \infty \right)
\]

So the correct response is:
**left parenthesis negative infinity comma negative 1 right parenthesis union left parenthesis 1 comma infinity right parenthesis**.