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The question is asking which of the given statements, if true, would be sufficient to conclude that there exists a number \( c \) in the interval \([-2, 2]\) such that \( g(c) = 6 \).

Let's analyze each statement using the Intermediate Value Theorem (IVT), which is often useful when reasoning about whether a function takes a particular value in an interval.

### Intermediate Value Theorem (IVT)
The IVT states that if a function \( g \) is continuous on a closed interval \([a, b]\), and \( g(a) \neq g(b) \), then for every value \( y \) between \( g(a) \) and \( g(b) \), there exists some \( c \in (a, b) \) such that \( g(c) = y \). 

In this problem, we are looking for \( c \) such that \( g(c) = 6 \) in the interval \([-2, 2]\). 

From the table, we see that:
- \( g(-1) = -1 \)
- \( g(0) = 4 \)
- \( g(1) = 11 \)

Since \( 6 \) is between \( g(0) = 4 \) and \( g(1) = 11 \), if \( g \) is continuous on the interval \([0, 1]\), the IVT guarantees that there is a \( c \in (0, 1) \) such that \( g(c) = 6 \).

Now, let's review each statement:

### Statement I: "g is defined for all \( x \) in the interval \([-2, 2]\)."
- This statement alone is not sufficient because simply knowing that \( g \) is defined doesn't imply that \( g(c) = 6 \). We need more information about the behavior of the function (e.g., continuity or increasing/decreasing nature) to apply the IVT.

### Statement II: "g is increasing on the interval \([-2, 2]\)."
- If \( g \) is increasing on the interval \([-2, 2]\), we can conclude that \( g \) takes every value between \( g(0) = 4 \) and \( g(1) = 11 \). Since \( 6 \) lies between these two values, this statement would be sufficient to conclude that there is a \( c \in (0, 1) \) such that \( g(c) = 6 \).

### Statement III: "g is continuous on the interval \([-2, 2]\)."
- If \( g \) is continuous on \([-2, 2]\), we can apply the Intermediate Value Theorem. Since \( g(0) = 4 \) and \( g(1) = 11 \), and 6 is between these values, continuity would guarantee the existence of a \( c \in (0, 1) \) such that \( g(c) = 6 \). Thus, this statement would also be sufficient.

### Conclusion:
- **Statement II (g is increasing)** and **Statement III (g is continuous)** are individually sufficient to conclude that there exists a number \( c \) such that \( g(c) = 6 \). 
- **Statement I (g is defined)** alone is not sufficient.

Would you like further details or have any questions?

### Related Questions:
1. How does the Intermediate Value Theorem apply to non-continuous functions?
2. What if the function was decreasing instead of increasing? How would that affect the conclusion?
3. Can we guarantee \( g(c) = 6 \) if only the endpoints of the interval are given and \( g \) is not continuous?
4. What is the difference between continuity and differentiability in ensuring the existence of solutions like this?
5. Could a function be defined but not continuous on an interval? How does that affect the Intermediate Value Theorem?

**Tip:** When using the IVT, always check whether the function is continuous over the interval and whether the value you're interested in lies between the function's values at the interval's endpoints.

The table above gives values of a function g at selected values of x. Which of the following statements, if true, would be individually sufficient to conclude that there exists a number c in the interval [-2, 2] such that g(c) = 6?

Understand how to use the Intermediate Value Theorem to conclude that there exists a value c in the interval [-2, 2] where g(c) = 6. This problem explores the application of continuity and monotonicity on function values. Suitable for students in Grades 11-12.

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