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Let's analyze the given functions and complete the parts (a) and (b) of the problem.

Given functions:
\[ f(x) = 3^x \]
\[ g(x) = 2x^3 \]

(a) Fill in the table below. Note that the table is already filled in for \( x = 3 \).

For \( x = 3 \):
\[ f(3) = 3^3 = 27 \]
\[ g(3) = 2 \cdot 3^3 = 2 \cdot 27 = 54 \]

Let's complete the table for the other given values of \( x \).

For \( x = 4 \):
\[ f(4) = 3^4 = 81 \]
\[ g(4) = 2 \cdot 4^3 = 2 \cdot 64 = 128 \]

For \( x = 5 \):
\[ f(5) = 3^5 = 243 \]
\[ g(5) = 2 \cdot 5^3 = 2 \cdot 125 = 250 \]

For \( x = 6 \):
\[ f(6) = 3^6 = 729 \]
\[ g(6) = 2 \cdot 6^3 = 2 \cdot 216 = 432 \]

For \( x = 7 \):
\[ f(7) = 3^7 = 2187 \]
\[ g(7) = 2 \cdot 7^3 = 2 \cdot 343 = 686 \]

Now we have the completed table:

| \( x \) | \( f(x) = 3^x \) | \( g(x) = 2x^3 \) |
|--------|------------------|-------------------|
| 3      | 27               | 54                |
| 4      | 81               | 128               |
| 5      | 243              | 250               |
| 6      | 729              | 432               |
| 7      | 2187             | 686               |

(b) For \( x \geq 4 \), the table suggests that \( f(x) \) is greater than \( g(x) \).

Would you like any further details or have any questions?

Here are 8 questions related to this problem:

1. How do you determine the value of \( 3^x \) for different \( x \) values?
2. How do you determine the value of \( 2x^3 \) for different \( x \) values?
3. Why does \( f(x) \) grow faster than \( g(x) \) for larger values of \( x \)?
4. What happens to \( f(x) \) and \( g(x) \) as \( x \) approaches infinity?
5. How can you use logarithms to compare the growth rates of \( f(x) \) and \( g(x) \)?
6. What other functions could you compare with similar methods?
7. How can these exponential and polynomial functions be applied in real-world contexts?
8. How would the table change if we used \( f(x) = 2^x \) and \( g(x) = x^3 \)?

**Tip:** When comparing functions, always consider their behavior as \( x \) increases, as this can reveal important differences in their growth rates.

Explore how exponential function f(x) = 3^x and polynomial function g(x) = 2x^3 compare in growth rates. Includes a detailed table analysis for x values from 3 to 7, demonstrating how f(x) grows faster than g(x) for x >= 4. Learn about logarithmic comparisons and real-world applications of these functions.

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