Let's analyze the equation \( m(p) = 20(2^p) \).
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Understanding the Equation:
- \( 20 \) is the initial amount of money.
- \( 2^p \) indicates that for each person who passes it on, the amount of money doubles.
- As \( p \) (the number of people who pass the money on) increases, \( m \) (the total amount of money) increases exponentially.
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Behavior of the Function:
- As \( p \) increases, \( m \) increases quickly (exponentially), meaning the amount of money grows rapidly as more people opt to pass it on.
- Thus, the statement "As p increases, m increases quickly" is true.
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Domain of the Function:
- Here, \( p \) can be any non-negative integer (0, 1, 2, ...), meaning \( p \) can take values greater than or equal to 0, so the domain is \( p \geq 0 \) and not \( p \geq 20 \).
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Range of the Function:
- The minimum amount when \( p = 0 \) is \( m(0) = 20(2^0) = 20 \).
- As \( p \) increases, \( m \) grows without bound. Therefore, the range of \( m \) is \( m \geq 20 \).
Based on this analysis, we can analyze the provided statements:
- "As p increases, m increases quickly at first and then slowly." – This is not true; it increases quickly and continues to do so.
- "The domain (p-values) include the values greater than or equal to 20 (p≥20)." – This is not true; the domain starts from 0, i.e., \( p \geq 0 \).
- "As p increases, m decreases slowly at first and then quickly." – This is not true; \( m \) does not decrease.
- "The range (m-values) includes the values (m≥20)." – This is true; the minimum value is 20, and it increases without bound.
Thus, the correct answer is:
The range (m-values) includes the values (m≥20).