Let's solve the given problems using the graph of the function $f(x)$.

(a) $f(-3)$

From the graph description, $f(x)$ crosses the x-axis at $x = -3$. This means that the value of $f(-3)$ is 0, because the x-axis represents the points where the function equals zero.

Thus, $f(-3) = 0$.

(b) $f(3)$

At $x = 3$, the graph transitions from a linear section to a curved section. The linear part ends with an open point at $(3, 6)$, but the second part begins at a closed point at $(3, -9)$. Since the closed point represents the actual value of the function at $x = 3$, we have:

Thus, $f(3) = -9$.

(c) $f(9)$

From the graph description, $f(x)$ passes through the point $(9, 9)$, so the value of $f(9)$ is simply the y-coordinate at that point.

Thus, $f(9) = 9$.

(d) $f(9) - f(3)$

We already found that $f(9) = 9$ and $f(3) = -9$, so:

$f(9) - f(3) = 9 - (-9) = 9 + 9 = 18$

(e) $\frac{f(9) - f(3)}{9 - 3}$

We calculated $f(9) - f(3) = 18$, and the denominator is $9 - 3 = 6$, so:

$\frac{f(9) - f(3)}{9 - 3} = \frac{18}{6} = 3$

Final Answers:

(a) $f(-3) = 0$
(b) $f(3) = -9$
(c) $f(9) = 9$
(d) $f(9) - f(3) = 18$
(e) $\frac{f(9) - f(3)}{9 - 3} = 3$

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Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. What is the slope of the linear part of the graph between points $(-9, -6)$ and $(-3, 0)$?
2. What would be the equation of the linear part of the function?
3. How would you describe the concavity of the curved portion of the graph?
4. What are the coordinates of the minimum and maximum points on the graph?
5. What is the average rate of change of the function over the interval $[3, 9]$?

Tip: Understanding function behavior from graphs often involves identifying key points (intercepts, peaks, troughs) and noting any changes in slope or curvature.