Let's analyze the problem from the image and address the questions step by step.

The graph represents a function $f(x)$ and has several features to analyze:

(a) Find the domain and the range of $f(x)$:

1. Domain: The domain is the set of all possible $x$-values for which the function is defined. From the graph, the function extends horizontally from $x = -4$ to $x = 4$, so: $\text{Domain: } [-4, 4]$

2. Range: The range is the set of all $y$-values that the function attains. From the graph, the lowest point is $y = -3$ and the highest point is $y = 3$, so: $\text{Range: } [-3, 3]$

(b) Intervals where $f(x)$ is increasing, decreasing, or constant:

• Increasing: The function is increasing where the slope of the curve is positive (going up). From the graph: $\text{Increasing: } (-4, -3), (-2, 2), (3, 4)$

• Decreasing: The function is decreasing where the slope is negative (going down). From the graph: $\text{Decreasing: } (-3, -2), (2, 3)$

• Constant: The function is constant where it has a horizontal segment (unchanging). From the graph: \text{Constant: } \text{No intervals where f(x) is constant.}

(c) Local minima and maxima:

1. Local Maximum: Points where the function transitions from increasing to decreasing:

   • At $x = -2$, $f(-2) = 1$.
   • At $x = 4$, $f(4) = 3$. $\text{Local Maxima: } (-2, 1), (4, 3)$

2. Local Minimum: Points where the function transitions from decreasing to increasing:

   • At $x = -3$, $f(-3) = 0$.
   • At $x = 2$, $f(2) = -1$. $\text{Local Minima: } (-3, 0), (2, -1)$

(d) Solve $f(x) > 0$:

Find intervals where $f(x) > 0$ (above the $x$-axis): $f(x) > 0 \text{ on: } (-4, -3) \cup (-2, 0) \cup (3, 4)$

(e) Solve $f(x) < 0$:

Find intervals where $f(x) < 0$ (below the $x$-axis): $f(x) < 0 \text{ on: } (-3, -2) \cup (0, 3)$

Let me know if you'd like to see any part explained in greater detail!

Related questions:

1. How do you determine critical points from a graph of $f(x)$?
2. How can you calculate the range algebraically for more complex functions?
3. What are the first derivative tests for identifying increasing and decreasing intervals?
4. What makes a point a local extremum, and how can you differentiate it from a global extremum?
5. How does solving $f(x) > 0$ relate to real-world applications?

Tip:

When analyzing a graph, always carefully observe transitions at key points (local maxima, minima, and intercepts) to accurately describe behavior!