https://file.aiquickdraw.com/mathbot/file/1726975938588erd5an6d.jpg

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!

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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.

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

Explore how to identify discontinuities, including jump discontinuities, and vertical asymptotes in the function g(x). This problem is designed for students in Grades 10-12, covering concepts of limits, asymptotes, and continuity.

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