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⋅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.
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Answer 1

To analyze the growth rates of the sequences for \( f(x) = 2 \cdot 2^x \) and \( g(x) = 10x + 2 \), let’s look closely at their behavior.

1. **Evaluate the difference between terms**:
- For \( f(x) = 2 \cdot 2^x \):
- \( f(1) = 4 \)
- \( f(2) = 8 \) (Difference = 4)
- \( f(3) = 16 \) (Difference = 8)
- \( f(4) = 32 \) (Difference = 16)
- \( f(5) = 64 \) (Difference = 32)
- \( f(6) = 128 \) (Difference = 64)
- \( f(7) = 256 \) (Difference = 128)
- \( f(8) = 512 \) (Difference = 256)
- \( f(9) = 1024 \) (Difference = 512)

- For \( g(x) = 10x + 2 \):
- \( g(1) = 12 \)
- \( g(2) = 22 \) (Difference = 10)
- \( g(3) = 32 \) (Difference = 10)
- \( g(4) = 42 \) (Difference = 10)
- \( g(5) = 52 \) (Difference = 10)
- \( g(6) = 62 \) (Difference = 10)
- \( g(7) = 72 \) (Difference = 10)
- \( g(8) = 82 \) (Difference = 10)
- \( g(9) = 92 \) (Difference = 10)

2. **Calculate the growth rates in intervals**:
- For \( f(x) \): The growth rates are doubling each time (4, 8, 16, ...).
- For \( g(x) \): The growth is linear with a constant difference of 10 for each subsequent term.

3. **Compare the sequences**:
- **Initially** (Term 1: \( f(1)=4 \) and \( g(1)=12 \)), \( g(x) \) is greater.
- **Between terms**:
- From Term 1 to Term 2 (4 to 8 for \( f(x) \), 12 to 22 for \( g(x) \)) shows \( g(x) \) is higher.
- By Term 3, \( f(3) = 16 \) and \( g(3) = 32 \), still \( g(x) \) is higher.
- At Term 4, \( f(4) = 32 \) and \( g(4) = 42 \), still \( g(x) \) is higher.
- At Term 5, \( f(5) = 64 \) vs \( g(5) = 52 \), here \( f(x) \) surpasses \( g(x) \).

4. **Conclusions about Statements**:
- **Statement #1**: True - \( f(x) \) eventually surpasses \( g(x) \).
- **Statement #2**: False - \( g(x) \) is always higher until Term 5.
- **Statement #3**: False - \( f(x) \) surpasses \( g(x) \), not the other way around.

Thus, the true statement is **Statement #1**.

**Final answer**: Statement #1 is true.