Final answer:
To sketch the desired function, draw a graph that increases from −∞ to −2, remains constant from −2 to −1, decreases from −1 to 3, and increases from 3 to ∞. Clearly label and mark the intervals on both axes using a numerical scale, and indicate the increasing and decreasing sections with upward and downward slopes, respectively.
Step-by-step explanation:
To sketch the graph of a function that behaves as specified, you will need to draw a graph that increases, decreases, and remains constant in different intervals. On the horizontal axis, which represents the input, label the points −∞ (negative infinity), −2, −1, 3, and ∞ (infinity). On the vertical axis, use an arbitrary numerical scale that clearly shows the increase and decrease of the function.
The function should:
- Increasing from −∞ to −2 means the function's value should be continuously getting higher. You might start low on the graph and slope upwards as you move towards −2.
- Constant from −2 to −1 means the graph should be a horizontal line for this interval. You can draw a flat line between these points.
- Decreasing from −1 to 3 indicates the function's value is getting lower. The graph should slope downwards from −1 ending lower at 3.
- Finally, increasing from 3 to ∞ means the function's value should rise again indefinitely past the point at 3.
The regions where the function is increasing, decreasing, and constant should be clearly marked on the graph, with arrows indicating that the function approaches infinity.