Let's work through this problem step by step.

Part (a): Completing the Table

For the given functions:

• $f(x) = 5^x$
• $g(x) = 3x^3$

We'll calculate each function's values for integer values of $x$ from 0 to 5.

Calculations:

1. When $x = 0$:

   • $f(0) = 5^0 = 1$
   • $g(0) = 3 \times 0^3 = 0$

2. When $x = 1$:

   • $f(1) = 5^1 = 5$
   • $g(1) = 3 \times 1^3 = 3$

3. When $x = 2$:

   • $f(2) = 5^2 = 25$
   • $g(2) = 3 \times 2^3 = 3 \times 8 = 24$

4. When $x = 3$:

   • $f(3) = 5^3 = 125$
   • $g(3) = 3 \times 3^3 = 3 \times 27 = 81$

5. When $x = 4$:

   • $f(4) = 5^4 = 625$
   • $g(4) = 3 \times 4^3 = 3 \times 64 = 192$

6. When $x = 5$:

   • $f(5) = 5^5 = 3125$
   • $g(5) = 3 \times 5^3 = 3 \times 125 = 375$

Table Completion:

Part (b): Sketching Both Functions

• To sketch these functions, you would plot the points obtained from the table for $f(x)$ and $g(x)$.
• Notice that $f(x) = 5^x$ grows exponentially, while $g(x) = 3x^3$ grows polynomially.
• Initially, $g(x)$ is close to or even larger than $f(x)$ for smaller values of $x$, but as $x$ increases, $f(x)$ grows much faster due to the nature of exponential functions.

Part (c): Dominance of One Function

Exponential functions grow faster than polynomial functions for large values of $x$. Thus, $f(x)$ will eventually dominate $g(x)$.

• Observing the table, $f(x)$ begins to dominate $g(x)$ starting from $x = 3$ and beyond.

So, the answer for (c) would be:

• $f(x)$ dominates over the interval $(3, \infty)$.

Would you like more details on any step, or do you have specific questions?

Here are some related questions to deepen your understanding:

1. Why does an exponential function eventually grow faster than a polynomial function?
2. How would the table values change if the base of $f(x)$ was smaller, such as $f(x) = 2^x$?
3. What happens to the dominance interval if $g(x)$ had a higher degree, say $g(x) = 3x^4$?
4. Can you find a general rule for when an exponential function dominates a polynomial function?
5. How would the answers change if $g(x)$ was a logarithmic function, like $g(x) = \ln(x)$?

Tip: Exponential functions of the form $a^x$, where $a > 1$, grow faster than any polynomial function as $x$ approaches infinity.