https://file.aiquickdraw.com/mathbot/file/1726707788606abpzmmoy.jpg

To solve the problem, we need to compare the values of \( f(2) \) and \( g(2) \), where the functions are defined as:

- \( f(x) = 4x^5 \)
- \( g(x) = 8 \cdot 4^x \)

### Step 1: Calculate \( f(2) \)
\[ f(x) = 4x^5 \]
Substitute \( x = 2 \):
\[
f(2) = 4 \cdot (2^5) = 4 \cdot 32 = 128
\]

### Step 2: Calculate \( g(2) \)
\[ g(x) = 8 \cdot 4^x \]
Substitute \( x = 2 \):
\[
g(2) = 8 \cdot 4^2 = 8 \cdot 16 = 128
\]

### Step 3: Compare \( f(2) \) and \( g(2) \)
Since \( f(2) = 128 \) and \( g(2) = 128 \), we conclude:
\[
f(2) = g(2)
\]

### Conclusion:
The correct answer is \( f(2) = g(2) \).

Would you like more details or explanations?

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Here are 5 related questions to extend your understanding:
1. How does the value of \( x \) affect the comparison of \( f(x) \) and \( g(x) \) for larger values?
2. How would the functions behave if the powers in \( f(x) \) or \( g(x) \) were changed?
3. What happens when \( x = 0 \) for both \( f(x) \) and \( g(x) \)?
4. How do exponential functions like \( g(x) = 8 \cdot 4^x \) differ from polynomial functions like \( f(x) = 4x^5 \)?
5. What happens if you substitute a negative value for \( x \) in both functions?

**Tip**: Always double-check your function evaluations, especially when working with powers and exponents.

Given the functions f(x) = 4x^5 and g(x) = 8 · 4^x, which of the following statements is true? f(2) < g(2), f(2) > g(2), or f(2) = g(2)?

Learn how to compare the values of a polynomial function f(x) = 4x^5 and an exponential function g(x) = 8 · 4^x at x = 2. A step-by-step solution to determine whether f(2) < g(2), f(2) > g(2), or f(2) = g(2). Suitable for students in Grades 9-12.

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