Let's break down this problem by interpreting the graph step by step. The key here is to identify the function's maximum and minimum values, the intervals where the function is increasing or decreasing, and the domain and range.

1. Maximum and Minimum

Looking at the graph:

• The maximum occurs where the function reaches its highest point. From the graph, it appears that the maximum is at $x = 1$ where the function reaches around $y = 4$.

• The minimum occurs where the function reaches its lowest point. From the graph, it appears that the minimum is at $x = -3$, where the function reaches around $y = -3$.

2. Intervals of Increasing and Decreasing

• The function is increasing where the slope is positive (the graph moves upwards as $x$ increases). This happens between $x = -3$ and $x = 1$.

• The function is decreasing where the slope is negative (the graph moves downwards as $x$ increases). The function decreases on two intervals:

  • From $x = -\infty$ to $x = -3$
  • From $x = 1$ onwards.

3. Domain and Range

• The domain of the function is all possible $x$-values the function covers. Since the graph continues to the left and right indefinitely, the domain is $(-\infty, \infty)$.

• The range of the function is the set of all possible $y$-values. The graph extends from $y = -3$ (the minimum) to $y = 4$ (the maximum), so the range is $[-3, 4]$.

Answers:

• Maximum at $x = 1$ with value 4.
• Minimum at $x = -3$ with value -3.
• Increasing on the interval $(-3, 1)$.
• Decreasing on the intervals $(-\infty, -3) \cup (1, \infty)$.
• Domain: $(-\infty, \infty)$.
• Range: $[-3, 4]$.

Would you like further clarification on any of these points?

Related Questions:

1. How do you determine the increasing and decreasing intervals from a graph?
2. What is the difference between local and absolute extrema?
3. How can you find the domain and range from a graph algebraically?
4. What are critical points, and how are they used in calculus?
5. How can you use derivatives to find increasing and decreasing intervals?

Tip:

When analyzing a graph, remember that increasing and decreasing intervals can always be checked by observing the slope or rate of change visually on the graph.