To analyze the problem from the provided image, let's interpret and solve it step-by-step:

Problem Breakdown:

1. Given Function: $f(x) = x$ is defined on the interval $[-3, 1]$, with $f : [-3, 1] \to \mathbb{R}$.
2. Tasks:
   • (a) Examine the sign of the function $f(x)$ over its domain.
   • (b) Analyze the increasing and decreasing nature of $f(x)$.

Solution:

(a) Sign of $f(x)$:

The function $f(x) = x$:

• Takes values equal to its input (identity function).
• On the interval $[-3, 1]$:
  • For $x \in [-3, 0)$, $f(x) < 0$ (negative values).
  • For $x = 0$, $f(x) = 0$ (neutral/zero).
  • For $x \in (0, 1]$, $f(x) > 0$ (positive values).

Conclusion:

• $f(x) < 0$ for $x \in [-3, 0)$,
• $f(x) = 0$ at $x = 0$,
• $f(x) > 0$ for $x \in (0, 1]$.

(b) Increasing and Decreasing Behavior:

The derivative of $f(x) = x$ is $f'(x) = 1$, which is constant and positive.

• Since $f'(x) > 0$ for all $x \in [-3, 1]$, the function is strictly increasing throughout its domain.

Conclusion:
The function $f(x) = x$ is increasing over the entire interval $[-3, 1]$.

Final Answer:

(a) Sign of $f(x)$:

• Negative on $[-3, 0)$,
• Zero at $x = 0$,
• Positive on $(0, 1]$.

(b) Increasing/Decreasing:

• $f(x) = x$ is strictly increasing on $[-3, 1]$.

Would you like additional explanations or visual graphs? Here are five related questions to deepen your understanding:

1. What happens to the sign of $f(x)$ if the domain changes to $[-3, 3]$?
2. How would the behavior of $f(x) = x^2$ differ in terms of sign and monotonicity?
3. Can you find an interval where the function $f(x) = x$ is decreasing?
4. What is the geometric meaning of the derivative being positive for $f(x) = x$?
5. How would a vertical shift, such as $g(x) = x + 2$, affect the sign and increasing/decreasing nature?

Tip: Always consider the derivative when analyzing increasing or decreasing behavior!