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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:
| \( x \) | \( f(x) \) | \( g(x) \) |
|---------|------------|------------|
| 0       | 1          | 0          |
| 1       | 5          | 3          |
| 2       | 25         | 24         |
| 3       | 125        | 81         |
| 4       | 625        | 192        |
| 5       | 3125       | 375        |

### 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 \( (3, \infty) \)**.

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

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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.

For each of the following functions f(x) = 5^x and g(x) = 3x^3, complete the table for integer values of x between 0 and 5. Then sketch both functions on the same grid and determine if one function eventually dominates. If so, state which function and at approximately what value of x.

Analyze the exponential function f(x) = 5^x and the polynomial function g(x) = 3x^3 by completing a value table and sketching both functions on a grid. Determine which function dominates as x increases and observe the nature of exponential versus polynomial growth. Suitable for high school students studying advanced algebra.

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