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

Given the functions:

- \( f(x) = 5x^4 \)
- \( g(x) = 6 \cdot 3^x \)

Let's analyze each of the statements that could be true based on the nature of these functions.

### 1. **\( f(x) \) is a polynomial function.**
- Yes, \( f(x) = 5x^4 \) is a polynomial of degree 4. Therefore, this statement is **true**.

### 2. **\( g(x) \) is an exponential function.**
- Yes, \( g(x) = 6 \cdot 3^x \) is an exponential function because the variable \( x \) is in the exponent of a constant base (3). Therefore, this statement is also **true**.

### 3. **\( f(x) \) grows faster than \( g(x) \) for large values of \( x \).**
- For large values of \( x \), exponential functions like \( g(x) = 6 \cdot 3^x \) tend to grow much faster than polynomial functions like \( f(x) = 5x^4 \). Therefore, this statement is **false**.

### 4. **\( f(x) \) and \( g(x) \) have the same growth rate.**
- As explained in the previous point, exponential functions grow faster than polynomial functions. Thus, they do not have the same growth rate. This statement is **false**.

### 5. **\( f(x) \) and \( g(x) \) intersect at some point.**
- To determine whether \( f(x) \) and \( g(x) \) intersect, we would need to solve the equation \( 5x^4 = 6 \cdot 3^x \). This may or may not have a solution, so without further analysis, we cannot definitively state if this is true or false without solving this equation numerically.

### Summary of True Statements:
- **Statement 1:** \( f(x) \) is a polynomial function.
- **Statement 2:** \( g(x) \) is an exponential function.

Would you like to go into more details or discuss any particular statement further?

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Here are 5 related questions to expand your understanding:

1. How do you differentiate between polynomial and exponential functions?
2. Why do exponential functions grow faster than polynomial functions as \( x \) becomes large?
3. How would you solve for the intersection of two functions numerically?
4. What is the general form of an exponential function, and how does its graph behave?
5. Can two functions with different growth rates intersect at more than one point?

**Tip:** Exponential functions tend to dominate polynomials for large values of \( x \) due to their rapid growth.

Given the functions f(x) = 5x^4 and g(x) = 6 ⋅ 3^x, which of the following statements is true?

Explore the properties of polynomial and exponential functions through the comparison of f(x) = 5x^4 and g(x) = 6 ⋅ 3^x. Learn about the growth rates of functions and determine the true statements about these two functions. Suitable for Grades 10-12.

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