The graph shows the function $y = f(x)$ with important points and features marked. Let's break down and solve each part of the problem based on the graph.

1. Find $f(2)$

Looking at the graph, when $x = 2$, the corresponding $y$-value (point on the graph) is $-1$. Therefore: $f(2) = -1$

2. Find all the $x$-values such that $f(x) = 0$

To find where $f(x) = 0$, we look for points where the graph crosses the x-axis. The graph intersects the x-axis at the following points: $x = -3, x = -1, x = 0, x = 3$ So, the $x$-values where $f(x) = 0$ are $-3, -1, 0, 3$.

3. Find the range of $f$

The range of a function is the set of possible $y$-values. From the graph, the function reaches a maximum at $y = 3$ (at $x = 4$) and a minimum at $y = -3$ (at $x = -4$). Therefore, the range of $f$ is: $\text{Range of } f = [-3, 3]$

4. Find all local extrema of $f$

Local extrema are the local minimum and maximum points. By observing the graph:

• There is a local minimum at $(-4, -3)$
• There is a local maximum at $(-1, 1)$
• There is a local minimum at $(2, -1)$

So, the local extrema are:

• Local minimum at $(-4, -3)$
• Local maximum at $(-1, 1)$
• Local minimum at $(2, -1)$

5. Find all absolute extrema of $f$

Absolute extrema are the highest and lowest points on the entire graph. From the graph:

• The absolute minimum is at $(-4, -3)$
• The absolute maximum is at $(4, 3)$

So, the absolute extrema are:

• Absolute minimum: $(-4, -3)$
• Absolute maximum: $(4, 3)$

Do you want more detailed explanations or have any specific questions on these results?

Here are some related questions to consider:

1. What happens to the function as $x \to -\infty$ based on the arrow at $x = -4$?
2. How do you determine if a point is a local maximum or minimum?
3. What is the difference between absolute and local extrema?
4. How does the closed endpoint at $(4, 3)$ influence the range of the function?
5. Can the function have additional extrema beyond what is shown in the graph?

Tip: When finding the range of a function, always check the endpoints and any maxima or minima within the interval.