Math Problem Statement

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?

Solution

Given the functions:

  • f(x)=5x4f(x) = 5x^4
  • g(x)=63xg(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)f(x) is a polynomial function.

  • Yes, f(x)=5x4f(x) = 5x^4 is a polynomial of degree 4. Therefore, this statement is true.

2. g(x)g(x) is an exponential function.

  • Yes, g(x)=63xg(x) = 6 \cdot 3^x is an exponential function because the variable xx is in the exponent of a constant base (3). Therefore, this statement is also true.

3. f(x)f(x) grows faster than g(x)g(x) for large values of xx.

  • For large values of xx, exponential functions like g(x)=63xg(x) = 6 \cdot 3^x tend to grow much faster than polynomial functions like f(x)=5x4f(x) = 5x^4. Therefore, this statement is false.

4. f(x)f(x) and g(x)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)f(x) and g(x)g(x) intersect at some point.

  • To determine whether f(x)f(x) and g(x)g(x) intersect, we would need to solve the equation 5x4=63x5x^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)f(x) is a polynomial function.
  • Statement 2: g(x)g(x) is an exponential function.

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


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 xx 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 xx due to their rapid growth.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Exponential Functions
Growth Rates of Functions

Formulas

f(x) = 5x^4
g(x) = 6 ⋅ 3^x

Theorems

Growth Comparison Theorem (exponential vs polynomial functions)

Suitable Grade Level

Grades 10-12