From the graph of $f(x)$ provided, let's analyze the domain and range.

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

The domain refers to all possible $x$-values for which the function is defined.

• On the leftmost part, the graph starts at $x = -6$, and there's an open circle at $x = -6$ (not included), and a closed circle at $x = -3$ (included).
• Moving to the middle section, the function continues from $x = 0$ to $x = 6$, with open circles at $x = 0$ and $x = 6$, meaning both are not included.
• There is another small horizontal segment from $x = -8$ to $x = -7$, with a closed circle on $x = -8$ and an open circle at $x = -7$.

So, the domain is:

$\text{Domain of } f(x) = [-8, -7) \cup (-6, -3] \cup (0, 6)$

(b) Range of $f(x)$:

The range refers to all possible $y$-values the function can take.

• The horizontal segment from $x = -8$ to $x = -7$ has $y = -4$, which is included in the range.
• The next visible $y$-values range from $y = 1$ to $y = 5$ (from $x = -6$ to $x = -3$), but with an open circle at $y = 5$.
• The "V" shape section starts from $y = 1$ to $y = 5$ (excluding 5 due to the open circles) and extends upwards without an upper bound.

So, the range is:

$\text{Range of } f(x) = [-4] \cup [1, 5)$

Final Answers:

• Domain of $f(x)$: $[-8, -7) \cup (-6, -3] \cup (0, 6)$
• Range of $f(x)$: $[-4] \cup [1, 5)$

Would you like any more details on this solution?

Here are 5 related questions for further practice:

1. How would the domain change if the open circles became closed?
2. How can the function be expressed piecewise given the graph?
3. What happens to the domain and range if the graph is shifted 2 units upwards?
4. How do you interpret vertical and horizontal shifts based on domain and range?
5. How would you determine the domain and range for a parabolic graph?

Tip: Remember that open circles indicate values that are not included in the domain or range, while closed circles show values that are included.