Let's analyze the graph and answer each part of the question.

(i) Domain of $g$ (in interval notation)

The domain represents the set of all possible $x$-values for which the function is defined.

• From the graph, the function seems to extend indefinitely to the left and right, covering all $x$-values except there is a gap at $x = 3$ where the function is undefined.

• Domain: $(-\infty, 3) \cup (3, \infty)$

(ii) Range of $g$ (in interval notation)

The range represents the set of all possible $y$-values that the function can take.

• The graph has $y$-values from $-4$ (the lowest point) up to 4 (the highest point).

• Range: $[-4, 4]$

(iii) $g(-2)$, $g(0)$, $g(2)$

From the graph:

• $g(-2) = -3$ (at $x = -2$, the $y$-value is $-3$).
• $g(0) = -2$ (at $x = 0$, the $y$-value is $-2$).
• $g(2) = 0$ (at $x = 2$, the $y$-value is $0$).

So,

• $g(-2) = -3$
• $g(0) = -2$
• $g(2) = 0$

(iv) All values of $x$ such that $g(x) = -4$

To find where $g(x) = -4$, we look at where the graph touches $y = -4$.

• From the graph, $g(x) = -4$ at $x = 1$.

So, $x = 1$.

(v) Intervals of increase

The function is increasing when the graph moves upward as $x$ increases.

• From $x = 1$ to $x = 3$
• From $x = 3$ to $\infty$

So, the intervals of increase are:

• $(1, 3)$ and $(3, \infty)$.

(vi) Intervals of decrease

The function is decreasing when the graph moves downward as $x$ increases.

• From $-\infty$ to $x = 1$

So, the interval of decrease is:

• $(-\infty, 1)$

(vii) Any local maximum and local minimum values

• Local minimum: Occurs at $x = 1$, where $g(1) = -4$.
• Local maximum: Occurs at $x = 3$, where $g(3) = 0$ (even though there's a hole, it's the local maximum of the continuous part).

Would you like further details on any part of the solution? Here are 5 follow-up questions to expand your understanding:

1. How would the domain change if the hole at $x = 3$ were filled?
2. What would be the domain if the function were not continuous at $x = 1$?
3. How do you identify intervals of increase and decrease from a graph?
4. What happens to the range if the graph is shifted vertically?
5. How do you find the local maxima and minima of a function from its graph?

Tip: Always check for discontinuities (like holes or jumps) when determining the domain of a function.