Let us solve the questions based on the graph provided in the image:

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

• Domain: The domain of $f(x)$ includes all $x$-values for which the function is defined. From the graph, the function is defined for $x$ in the interval: $[-4, 4]$

• Range: The range is the set of all $y$-values that the function takes. From the graph, the $y$-values range from $-3$ to $3$, so the range is: $[-3, 3]$

(b) The intervals on which $f$ is increasing, decreasing, or constant:

• Increasing: The graph is increasing where $f(x)$ moves upward as $x$ increases. $(-3, -2) \quad \text{and} \quad (3, 4)$

• Decreasing: The graph is decreasing where $f(x)$ moves downward as $x$ increases. $(-4, -3) \quad \text{and} \quad (-2, 3)$

• Constant: There are no intervals where the function is constant.

(c) Find the local minimum and the local maximum:

• Local Minimums:

  • At $x = -3$, $f(-3) = 0$
  • At $x = 2$, $f(2) = -1$

• Local Maximums:

  • At $x = -2$, $f(-2) = 1$
  • At $x = 4$, $f(4) = 3$

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

Find where $f(x)$ is above the $x$-axis: $(-3, -2) \quad \text{and} \quad (3, 4)$

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

Find where $f(x)$ is below the $x$-axis: $(-4, -3) \quad \text{and} \quad (-2, 3)$

Would you like a detailed explanation for any specific part or have any additional questions?

Here are 5 related questions for deeper exploration:

1. What is the significance of local extrema in real-world applications?
2. Can a function have multiple ranges for different subdomains?
3. How do we determine intervals of increase or decrease analytically without a graph?
4. What are the conditions for a function to have no constant intervals?
5. How can the concept of $f(x) > 0$ and $f(x) < 0$ be extended to inequalities involving $f(x) \geq 0$?

Tip: To better analyze graphs, always check for points of interest like roots, turning points, and behavior at boundaries.