Given the functions f, of, x, equals, 5, x, to the power 4*f*(*x*)=5*x*4 and g, of, x, equals, 6, dot, 3, to the power x*g*(*x*)=6⋅3*x*, which of the following statements is true?

Given the functions \( f(x) = 5x^4 \) and \( g(x) = 6 \cdot 3^x \), let's analyze the given options:

### Understanding the Functions:
1. **\( f(x) = 5x^4 \)**:
   - This is a polynomial function of degree 4. As \( x \) increases, \( f(x) \) grows rapidly, particularly for large positive or negative values of \( x \).
   
2. **\( g(x) = 6 \cdot 3^x \)**:
   - This is an exponential function with base 3. Exponential functions grow much faster than polynomial functions as \( x \) increases.

### Comparing Growth Rates:
- **Exponential functions** like \( g(x) = 6 \cdot 3^x \) tend to grow faster than **polynomial functions** like \( f(x) = 5x^4 \) as \( x \) becomes large.

### Possible True Statements:
1. \( f(x) > g(x) \) for all \( x \).
2. \( f(x) < g(x) \) for all \( x \).
3. \( f(x) = g(x) \) for some \( x \).
4. \( f(x) = g(x) \) for all \( x \).
5. \( f(x) > g(x) \) for large \( x \).
6. \( g(x) > f(x) \) for large \( x \).

### Evaluation:
- For large values of \( x \), \( g(x) = 6 \cdot 3^x \) will eventually outgrow \( f(x) = 5x^4 \), because exponential growth outpaces polynomial growth.
  
Therefore, **the true statement is**:

- **\( g(x) > f(x) \) for large \( x \).**

Would you like more details on this comparison or have any questions? Here are five related questions:

1. How do polynomial and exponential functions compare in growth rate?
2. What is the general form of an exponential function?
3. Can a polynomial ever outgrow an exponential function for any value of \( x \)?
4. What happens to \( f(x) \) and \( g(x) \) as \( x \) approaches infinity?
5. How can you determine when \( f(x) = g(x) \) for specific values of \( x \)?

**Tip:** When comparing different types of functions, always consider their long-term behavior, particularly as \( x \) approaches infinity.

Explore the comparison between polynomial function f(x) = 5x^4 and exponential function g(x) = 6 ⋅ 3^x. Understand their growth rates and learn why exponential functions outpace polynomial functions for large values of x. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation