The image shows a graph of the function $g(x)$ with questions regarding its behavior at different values of $x = a$.

Let's analyze the questions.

Question 1:

"For which value(s) of $a$ does $g(x)$ approach a different number from the right side than it approaches from the left side?"

This is asking about discontinuities, specifically jump discontinuities. Based on the graph:

• At $a = 0$, there is a jump discontinuity where the limit from the left and right are different.
• At $a = 1$, it seems like there's a removable discontinuity or a hole, but the function approaches the same value from both sides.
• At $a = 3$, the graph shows a continuous curve.

So, the correct value for this is:

• $a = 0$

Question 2:

"For which value(s) of $a$ does $g(x)$ increase or decrease without bound as $x$ approaches $a$, or for which value(s) of $a$ is $g(x)$ not defined?"

This is asking where the function either has a vertical asymptote (where the function goes to infinity) or where it's not defined.

• At $a = -2$ and $a = -1$, the graph shows vertical asymptotes where the function increases or decreases without bound.
• At $a = 0$, there's a hole in the graph, meaning the function is not defined there.

So, the correct values for this are:

• $a = -2$, $a = -1$, and $a = 0$.

Let me know if you'd like more detailed explanations or have further questions!

Here are five related questions to deepen your understanding:

1. What is the difference between a jump discontinuity and a removable discontinuity?
2. How can we tell if a function has a vertical asymptote from its graph?
3. What does it mean for a function to be continuous at a point?
4. What are the formal definitions of left-hand and right-hand limits?
5. How can we algebraically verify if a function is not defined at a certain point?

Tip: When examining graphs for discontinuities, always check for differences in left-hand and right-hand limits at critical points.