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.