Use the table to answer the question.

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 evaluate the truth of the statements regarding the growth rates of the sequences for \( f(x) = 2 \cdot 2^x \) and \( g(x) = 10x + 2 \), let's analyze the sequences and their growth rates.

The sequence for \( f(x) \) is:

• 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 sequence for \( g(x) \) is:

• 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

Now, let's compare their growth rates:

For \( f(x) \):

• Growth from Term 1 to Term 2: 8 - 4 = 4
• Growth from Term 2 to Term 3: 16 - 8 = 8
• Growth from Term 3 to Term 4: 32 - 16 = 16
• Growth from Term 4 to Term 5: 64 - 32 = 32
• Growth from Term 5 to Term 6: 128 - 64 = 64
• Growth from Term 6 to Term 7: 256 - 128 = 128
• Growth from Term 7 to Term 8: 512 - 256 = 256
• Growth from Term 8 to Term 9: 1024 - 512 = 512

For \( g(x) \):

• Growth from Term 1 to Term 2: 22 - 12 = 10
• Growth from Term 2 to Term 3: 32 - 22 = 10
• Growth from Term 3 to Term 4: 42 - 32 = 10
• Growth from Term 4 to Term 5: 52 - 42 = 10
• Growth from Term 5 to Term 6: 62 - 52 = 10
• Growth from Term 6 to Term 7: 72 - 62 = 10
• Growth from Term 7 to Term 8: 82 - 72 = 10
• Growth from Term 8 to Term 9: 92 - 82 = 10

Analyze 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 false because the initial growth of \( g(x) \) is \( 10 \), while the initial growth of \( f(x) \) is \( 4 \). However, \( f(x) \) does increase and eventually surpasses \( g(x) \) in growth rate.

• 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."

  • Between the second term (growth of \( f(x) = 8 - 4 = 4 \)) and the fifth term (growth of \( f(x) = 64 - 32 = 32 \)), the growth of \( g(x) \) remains consistent at \( 10 \) and thus is greater than \( f(x) \) up until Term 4 (where \( f(x) = 16 \) and \( g(x) = 42 \)). This implies that Statement #2 is true.

• 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. The growth rate of \( f(x) \) eventually surpasses that of \( g(x) \), not the other way around.

Conclusion

The true statement is 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."

Answer 2

Use the image to answer the question.

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it.

The graph of f(x)=100⋅1.5x
and g(x)=100x+100
is shown. Which of the following statements is true?

Statement #1: 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)
.

Statement #2: The growth rate of g(x)
is greater than the growth rate of f(x)
between approximately x=0
and x=4
.

Statement #3: 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, by x=2.5
, surpasses the growth rate of g(x)
.

(1 point)
Statement #
is true.
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Answer 3

To evaluate the truth of the statements about the growth rates of \( f(x) = 100 \cdot 1.5^x \) and \( g(x) = 100x + 100 \), let's analyze their behavior and growth rates based on their mathematical forms.

1. Growth Behavior Analysis:

   • Exponential Function \( f(x) \):

     • As \( x \) increases, \( f(x) \) will grow exponentially because of the \( 1.5^x \) term. This means its growth rate will accelerate over time.

   • Linear Function \( g(x) \):

     • The function \( g(x) \) grows linearly at a constant rate. The rate of change (slope) is constant and equal to 100.

2. Initial Growth Rates:

   • At \( x = 0 \):
     • \( f(0) = 100 \cdot 1.5^0 = 100 \)
     • \( g(0) = 100 \cdot 0 + 100 = 100 \)

   The initial growth rate of both functions is the same.

3. Rate of Change Calculations:

   • The rate of growth for \( f(x) \) can be determined by calculating its derivative:
     • \( f'(x) = 100 \cdot 1.5^x \cdot \ln(1.5) \), which increases exponentially as \( x \) increases due to \( 1.5^x \).
   • The rate of growth for \( g(x) \) remains constant at \( g'(x) = 100 \).

Analyze the Statements

• Statement #1: "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. The growth rate of \( f(x) \) does become greater than that of \( g(x) \) at some point due to the exponential nature of \( f(x) \), and \( g(x) \)’s rate does not increase, but remains constant.

• Statement #2: "The growth rate of \( g(x) \) is greater than the growth rate of \( f(x) \) between approximately \( x=0 \) and \( x=4 \)."

  • This statement is likely false as well. Given that \( f(x) \) starts growing exponentially, although they may start at the same point at \( x=0 \), \( f(x) \) will eventually surpass \( g(x) \) as \( x \) increases.

• Statement #3: "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, by \( x=2.5 \), surpasses the growth rate of \( g(x) \)."

  • This statement is false as the growth rates of \( f(x) \) and \( g(x) \) initially start equal.

Conclusion

Given the analysis of the three statements, they all contain inaccuracies regarding the growth comparisons at the specified intervals according to the functions' behavior. Therefore, it seems that none of the statements are true based on the information provided.

If you have a graph that shows the two functions more clearly, it might help pinpoint exact crossover points more accurately. However, without visual confirmation, based on mathematical analysis: none of the statements are true.

If you have an option to select a "most correct" statement, you may need more context or precise points on the graph to determine if one statement seems more plausible than others. Otherwise, based on calculations, none are true.