Question

Use the table to answer the 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 𝑓 ( 𝑥 ) = 2 ⋅ 2 𝑥 4 8 16 32 64 128 256 512 1024 Sequence for g(x)=10x+2 𝑔 ( 𝑥 ) = 10 𝑥 + 2 12 22 32 42 52 62 72 82 92 The sequences for f(x)=2⋅2x 𝑓 ( 𝑥 ) = 2 ⋅ 2 𝑥 and g(x)=10x+2 𝑔 ( 𝑥 ) = 10 𝑥 + 2 are shown. Which of the following statements is true? Statement #1: While the growth rate of g(x) 𝑔 ( 𝑥 ) is initially greater than the growth rate of f(x) 𝑓 ( 𝑥 ) , the growth rate of f(x) 𝑓 ( 𝑥 ) keeps increasing and eventually surpasses the growth rate of g(x) 𝑔 ( 𝑥 ) . Statement #2: The growth rate of g(x) 𝑔 ( 𝑥 ) is greater than the growth rate of f(x) 𝑓 ( 𝑥 ) between the second term and the fifth term. Statement #3: While the growth rate of f(x) 𝑓 ( 𝑥 ) is initially greater than the growth rate of g(x) 𝑔 ( 𝑥 ) , the growth rate of g(x) 𝑔 ( 𝑥 ) keeps increasing and eventually surpasses the growth rate of f(x) 𝑓 ( 𝑥 ) . (1 point) Statement # is true.

Answer 1

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:

- **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)

- **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)

- **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)

- **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)

- **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.**