Question

Use the table to answer the question. 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⋅2^x 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 analyze the two sequences, we'll look at the values given for \( f(x) = 2 \cdot 2^x \) and \( g(x) = 10x + 2 \) at each term:

• For \( f(x) \):

  • \( f(1) = 4 \)
  • \( f(2) = 8 \)
  • \( f(3) = 16 \)
  • \( f(4) = 32 \)
  • \( f(5) = 64 \)
  • \( f(6) = 128 \)
  • \( f(7) = 256 \)
  • \( f(8) = 512 \)
  • \( f(9) = 1024 \)

• For \( g(x) \):

  • \( g(1) = 12 \)
  • \( g(2) = 22 \)
  • \( g(3) = 32 \)
  • \( g(4) = 42 \)
  • \( g(5) = 52 \)
  • \( g(6) = 62 \)
  • \( g(7) = 72 \)
  • \( g(8) = 82 \)
  • \( g(9) = 92 \)

Now, let's examine the growth rates of these sequences.

Growth Rates

• For \( f(x) \): The values of \( f(x) \) are:

  • \( f(1) = 4 \)
  • \( f(2) = 8 \): Increase of 4
  • \( f(3) = 16 \): Increase of 8
  • \( f(4) = 32 \): Increase of 16
  • \( f(5) = 64 \): Increase of 32
  • Continues to double each time.

• For \( g(x) \): The values of \( g(x) \) are:

  • \( g(1) = 12 \)
  • \( g(2) = 22 \): Increase of 10
  • \( g(3) = 32 \): Increase of 10
  • \( g(4) = 42 \): Increase of 10
  • \( g(5) = 52 \): Increase of 10
  • Remains linear with a constant increase of 10.

Analysis of Each Statement

• 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 \( g(x) \)."

  • Incorrect, since \( g(x) \) starts higher than \( f(x) \) at the first term, but \( g(x) \) never exceeds \( f(x) \) in successive terms.

• 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."

  • Incorrect, since while \( g(x) \) has higher values at first, the increases for \( g(x) \) are constant at 10, while \( f(x) \) increases grow exponentially (doubles).

• 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 \( f(x) \)."

  • Incorrect as the growth rate of \( g(x) \) is constant, while \( f(x) \) increases exponentially.

From the analysis above, none of the statements can be considered true because:

• \( f(x) \) starts from a lower value and grows exponentially, while \( g(x) \) grows linearly and does not surpass \( f(x) \) at any point in the given sequences.

Thus, none of the provided statements are true.