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Let's analyze and complete the table for the two functions given:

- \( f(x) = 4^x \)
- \( g(x) = x + 8 \)

The table values need to be calculated for \( x = 1, 2, 3, 4 \).

### For \( f(x) = 4^x \):
1. \( f(1) = 4^1 = 4 \)
2. \( f(2) = 4^2 = 16 \)
3. \( f(3) = 4^3 = 64 \)
4. \( f(4) = 4^4 = 256 \)

### For \( g(x) = x + 8 \):
1. \( g(1) = 1 + 8 = 9 \)
2. \( g(2) = 2 + 8 = 10 \)
3. \( g(3) = 3 + 8 = 11 \)
4. \( g(4) = 4 + 8 = 12 \)

### Completed Table:

| x  | \( f(x) = 4^x \) | \( g(x) = x + 8 \) |
|----|------------------|--------------------|
| 1  | 4                | 9                  |
| 2  | 16               | 10                 |
| 3  | 64               | 11                 |
| 4  | 256              | 12                 |

### Which function eventually exceeds the other?

While \( g(x) \) initially grows faster than \( f(x) \), after \( x = 2 \), \( f(x) \) starts to exceed \( g(x) \). For large values of \( x \), \( f(x) = 4^x \) will grow exponentially faster than the linear function \( g(x) = x + 8 \).

Let me know if you'd like further clarification or details!

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### Five follow-up questions:
1. What is the general difference between exponential and linear functions?
2. How would the table change if we evaluated \( x = 0 \) for both functions?
3. What happens to \( f(x) \) and \( g(x) \) as \( x \) becomes very large?
4. Can you compare the rate of growth of \( 4^x \) to that of other exponential functions, like \( 2^x \)?
5. What is the significance of the intersection point between the two functions?

**Tip:** Exponential functions like \( 4^x \) grow much faster than linear functions like \( x + 8 \) after a certain point.

Complete the table of values for f(x) = 4^x and g(x) = x + 8. Both f(x) and g(x) grow as x gets larger and larger. Which function eventually exceeds the other?

Learn how to complete a table of values for the exponential function f(x) = 4^x and the linear function g(x) = x + 8. This problem explores how exponential and linear functions grow differently and which eventually exceeds the other. Suitable for students in Grades 7-9.

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