Let's analyze the graph provided step by step to answer each of the questions about the function $f(x)$.

(a) Domain of $f$

The domain is the set of all possible input values (x-values) for which the function is defined. From the graph, the function starts at $x = -8$ and continues to $x = 4$. Therefore, the domain is: $\text{Domain: } [-8, 4]$

(b) Range of $f$

The range is the set of all possible output values (y-values). From the graph, the function has a maximum value of $y = 4$ and goes down to $y = -5$. Therefore, the range is: $\text{Range: } [-5, 4]$

(c) Zeros of $f$

Zeros are the x-values where $f(x) = 0$ (i.e., where the graph crosses the x-axis). From the graph, the zeros occur at $x = -6$ and $x = 2$. $\text{Zeros: } x = -6, 2$

(d) $f(0)$

To find $f(0)$, locate where $x = 0$ on the graph and find the corresponding $y$-value. From the graph: $f(0) = 2$

(e) Intervals where $f$ is increasing

The function is increasing where the graph is going upward. This occurs from $x = -8$ to $x = -4$ (since the graph rises from left to right in this interval). $\text{Increasing on: } [-8, -4]$

(f) Intervals where $f$ is decreasing

The function is decreasing where the graph is going downward. This occurs from $x = -4$ to $x = 4$. $\text{Decreasing on: } [-4, 4]$

(g) Values where $f(x) \leq 0$

The values of $x$ where the graph is below or on the x-axis (i.e., $f(x) \leq 0$) are from $x = -8$ to $x = -6$, and from $x = 2$ to $x = 4$. [ f(x) \leq 0 \text{ for } x \in [-8, -6] \cup [2, 4] ]

(h) Relative maxima or minima

• Relative maximum: The highest point within a certain interval. The relative maximum occurs at $x = -4$, where $f(x) = 4$.
• Relative minimum: There is no relative minimum, as the function does not have any lower turning points before continuing downward.

(i) Values of $x$ for which $f(x) = 4$

The value of $f(x) = 4$ occurs at $x = -4$. $f(x) = 4 \text{ at } x = -4$

(j) Is $f(-2)$ positive or negative?

To find $f(-2)$, locate $x = -2$ on the graph and find the corresponding $y$-value. From the graph, $f(-2)$ is positive because the graph is above the x-axis at $x = -2$. $f(-2) \text{ is positive.}$

Would you like further details or explanations? Here are some related questions to consider:

1. How do you determine the relative maxima and minima of a function from its graph?
2. What is the difference between absolute maxima/minima and relative maxima/minima?
3. How can the intervals of increase and decrease be found using the derivative of a function?
4. How would you describe the behavior of the function at $x = 4$ where the function terminates?
5. Can a function have more than one relative maximum?

Tip: A function's domain refers to the x-values for which the function is defined, and the range refers to the y-values the function can take. Understanding this distinction is key when interpreting graphs.