To analyze which statement is true, let’s first look at the growth rates of the two functions and their sequences.
For \( f(x) = 2 \cdot 2^x \):
- 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
The growth from term to term is:
- 4 → 8 (increase by 4)
- 8 → 16 (increase by 8)
- 16 → 32 (increase by 16)
- 32 → 64 (increase by 32)
- 64 → 128 (increase by 64)
- 128 → 256 (increase by 128)
- 256 → 512 (increase by 256)
- 512 → 1024 (increase by 512)
The growth is doubling each time, indicating exponential growth.
For \( g(x) = 10x + 2 \):
- 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
The growth from term to term is:
- 12 → 22 (increase by 10)
- 22 → 32 (increase by 10)
- 32 → 42 (increase by 10)
- 42 → 52 (increase by 10)
- 52 → 62 (increase by 10)
- 62 → 72 (increase by 10)
- 72 → 82 (increase by 10)
- 82 → 92 (increase by 10)
The growth rate of \( g(x) \) is constant (always increasing by 10).
Now, let's evaluate the statements:
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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 \( g(x) \).
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Initially, \( g(x) \) increases by 10 (second term grows at 10) while \( f(x) \) increases by only 4, so \( g(x) \) is greater at first. As \( f(x) \) grows, its growth rate eventually surpasses \( g(x) \)'s constant growth rate.
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Thus, this statement is true.
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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.
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The growth increments for \( f(x) \) during those terms are: 8, 16, 32 (increases by 8, then 16, then 32).
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For \( g(x) \), it grows by 10 each time.
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However, after the third term, the growth rate of \( f(x) \) surpasses \( g(x) \).
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Therefore, this statement is false.
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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 incorrect because \( g(x) \) has a constant growth rate and does not increase.
Hence, statement #1 is the only true statement.
The correct answer is:
Statement #1 is true.
1 answer