Use the table to answer the question.

Question

This table compares the sequence for two given functions for fixed term numbers (1-9). Term Number 1 2 3 4 5 6 7 8 9
Sequence for
f(x)=2⋅2x
4 8 16 32 64 128 256 512 1024
Sequence for
g(x)=10x+2
12 22 32 42 52 62 72 82 92

The sequences for f(x)=2⋅2x
and g(x)=10x+2

are shown. Which of the following statements is true?

Statement #1: While the growth rate of g(x)
is initially greater than the growth rate of f(x), the growth rate of f(x) keeps increasing and eventually surpasses the growth rate of g(x)

.

Statement #2: The growth rate of g(x)
is greater than the growth rate of f(x)

between the second term and the fifth term.

Statement #3: While the growth rate of f(x)
is initially greater than the growth rate of g(x), the growth rate of g(x) keeps increasing and eventually surpasses the growth rate of f(x)

.
(1 point)
Statement #
is true.

Answer 1

To determine which statement is true, we need to analyze the growth rates of the two sequences for each function.

1. Function f(x) = 2⋅2^x:

   • The values for f(x) are:
   • Term 1: 4
   • Term 2: 8
   • Term 3: 16
   • Term 4: 32
   • Term 5: 64
   • Term 6: 128
   • Term 7: 256
   • Term 8: 512
   • Term 9: 1024

   Growth from term to term:

   • 4 to 8: increase of 4
   • 8 to 16: increase of 8
   • 16 to 32: increase of 16
   • 32 to 64: increase of 32
   • 64 to 128: increase of 64
   • 128 to 256: increase of 128
   • 256 to 512: increase of 256
   • 512 to 1024: increase of 512

   The growth for f(x) doubles each time.

2. Function g(x) = 10x + 2:

   • The values for g(x) are:
   • Term 1: 12
   • Term 2: 22
   • Term 3: 32
   • Term 4: 42
   • Term 5: 52
   • Term 6: 62
   • Term 7: 72
   • Term 8: 82
   • Term 9: 92

   Growth from term to term:

   • 12 to 22: increase of 10
   • 22 to 32: increase of 10
   • 32 to 42: increase of 10
   • 42 to 52: increase of 10
   • 52 to 62: increase of 10
   • 62 to 72: increase of 10
   • 72 to 82: increase of 10
   • 82 to 92: increase of 10

   The growth for g(x) is constant at 10.

Now, let’s evaluate each statement:

1. Statement #1: "While the growth rate of g(x) is initially greater than the growth rate of f(x), the growth rate of f(x) keeps increasing and eventually surpasses the growth rate of g(x)."

   • The initial growth rate of g(x) (10) is less than the growth rate of f(x) (4). Thus, this statement is false.

2. Statement #2: "The growth rate of g(x) is greater than the growth rate of f(x) between the second term and the fifth term."

   • Between terms 2-5:
     • f(2) = 8 to f(5) = 64 shows increases of 4, 8, 16, and 32—last difference is 32.
     • g(2) = 22 to g(5) = 52 is constant at 10 for each step.
   • It is clear that f(x) surpasses g(x) for all terms beyond the second. This statement is false.

3. Statement #3: "While the growth rate of f(x) is initially greater than the growth rate of g(x), the growth rate of g(x) keeps increasing and eventually surpasses the growth rate of f(x)."

   • This is false. The growth rate of g(x) is constant at 10, while f(x) increases significantly more in a compounding manner.

Since all the statements are false based on our analysis, we can conclude:

None of the statements are true.

However, based on the provided statements, it seems there was a misunderstanding in their evaluation. The most applicable conclusion is that Statement #1 is the closest to be true, misunderstood slightly due to initial comparisons. In the event we're choosing the least incorrect, it would lead towards that conclusion.