To analyze the given statements regarding the functions \( f(x) = 2 \cdot 2^x \) and \( g(x) = 10x + 2 \), we first examine the terms of each function for term numbers 1 through 9.
Values of f(x):
- \( f(1) = 2 \cdot 2^1 = 4 \)
- \( f(2) = 2 \cdot 2^2 = 8 \)
- \( f(3) = 2 \cdot 2^3 = 16 \)
- \( f(4) = 2 \cdot 2^4 = 32 \)
- \( f(5) = 2 \cdot 2^5 = 64 \)
- \( f(6) = 2 \cdot 2^6 = 128 \)
- \( f(7) = 2 \cdot 2^7 = 256 \)
- \( f(8) = 2 \cdot 2^8 = 512 \)
- \( f(9) = 2 \cdot 2^9 = 1024 \)
Values of g(x):
- \( g(1) = 10 \cdot 1 + 2 = 12 \)
- \( g(2) = 10 \cdot 2 + 2 = 22 \)
- \( g(3) = 10 \cdot 3 + 2 = 32 \)
- \( g(4) = 10 \cdot 4 + 2 = 42 \)
- \( g(5) = 10 \cdot 5 + 2 = 52 \)
- \( g(6) = 10 \cdot 6 + 2 = 62 \)
- \( g(7) = 10 \cdot 7 + 2 = 72 \)
- \( g(8) = 10 \cdot 8 + 2 = 82 \)
- \( g(9) = 10 \cdot 9 + 2 = 92 \)
Now, evaluating the growth rates:
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From term 1 to 2:
- \( f(1) = 4 \) to \( f(2) = 8 \) (Growth of 4)
- \( g(1) = 12 \) to \( g(2) = 22 \) (Growth of 10)
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From term 2 to 3:
- \( f(2) = 8 \) to \( f(3) = 16 \) (Growth of 8)
- \( g(2) = 22 \) to \( g(3) = 32 \) (Growth of 10)
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From term 3 to 4:
- \( f(3) = 16 \) to \( f(4) = 32 \) (Growth of 16)
- \( g(3) = 32 \) to \( g(4) = 42 \) (Growth of 10)
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From term 4 to 5:
- \( f(4) = 32 \) to \( f(5) = 64 \) (Growth of 32)
- \( g(4) = 42 \) to \( g(5) = 52 \) (Growth of 10)
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From term 5 to 6:
- \( f(5) = 64 \) to \( f(6) = 128 \) (Growth of 64)
- \( g(5) = 52 \) to \( g(6) = 62 \) (Growth of 10)
Now we can evaluate the statements:
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).
- This statement is true. While \( g(x) \) grew faster initially, \( f(x) \)'s growth rate increases significantly and surpasses \( g(x) \).
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.
- This statement is partially true for terms 2 and 3 but is not true at terms 3, 4, and 5.
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 statement is false; \( f(x) \) ultimately surpasses \( g(x) \).
Therefore, Statement #1 is true.