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?

Answer

Multiple Choice Answers

f, of, 9, equals, g, of, 9*f*(9)=*g*(9)

f, of, 9, is less than, g, of, 9*f*(9)<*g*(9)

f, of, 9, is greater than, g, of, 9*f*(9)>*g*(9)




Let's calculate \( f(9) \) and \( g(9) \) using the given functions to determine the correct answer.

### Given Functions:
1. \( f(x) = 5x^4 \)
2. \( g(x) = 6 \cdot 3^x \)

### Step 1: Calculate \( f(9) \)
\[
f(9) = 5 \cdot 9^4
\]
\[
9^4 = 6561
\]
\[
f(9) = 5 \cdot 6561 = 32805
\]

### Step 2: Calculate \( g(9) \)
\[
g(9) = 6 \cdot 3^9
\]
\[
3^9 = 19683
\]
\[
g(9) = 6 \cdot 19683 = 118098
\]

### Step 3: Compare \( f(9) \) and \( g(9) \)
- \( f(9) = 32805 \)
- \( g(9) = 118098 \)

Since \( 32805 < 118098 \), the correct statement is:

\[
f(9) < g(9)
\]

### Final Answer:
**\( f(9) < g(9) \)** is the correct statement.

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

1. How do you determine the growth rate of different types of functions?
2. What is the significance of the base in exponential functions?
3. How would the comparison change if \( x \) were smaller?
4. Can you find other values of \( x \) where \( f(x) \) and \( g(x) \) might be equal?
5. What does the calculation tell you about the nature of polynomial vs. exponential functions?

**Tip:** When comparing functions, it helps to evaluate them at specific points to understand their relative sizes.

Explore the comparison of exponential function g(x) = 6 * 3^x and polynomial function f(x) = 5x^4 by evaluating f(9) and g(9). Understand how these functions behave differently at specific points. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation