To find the domain and range for the function \( p(n) \), which represents the total number of minutes Elena practiced in a week based on the number of practice days \( n \):
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Domain: Since Elena practices for 30 minutes each day, the possible number of practice days \( n \) is bounded by the maximum number of days in a week, which is 7. Thus, \( n \) can be any whole number from 0 (not practicing at all) to 7 (practicing every day):
\[ \text{Domain: } n \in {0, 1, 2, 3, 4, 5, 6, 7} \]
In interval notation, the domain can be expressed as: \[ \text{Domain: } [0, 7] \]
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Range: To calculate the range, we need to determine the total practice minutes \( p(n) \) for each possible value of \( n \): \[ p(n) = 30n \] If \( n = 0 \), then \( p(0) = 30 \cdot 0 = 0 \) minutes. If \( n = 1 \), then \( p(1) = 30 \cdot 1 = 30 \) minutes. If \( n = 2 \), then \( p(2) = 30 \cdot 2 = 60 \) minutes. If \( n = 3 \), then \( p(3) = 30 \cdot 3 = 90 \) minutes. If \( n = 4 \), then \( p(4) = 30 \cdot 4 = 120 \) minutes. If \( n = 5 \), then \( p(5) = 30 \cdot 5 = 150 \) minutes. If \( n = 6 \), then \( p(6) = 30 \cdot 6 = 180 \) minutes. If \( n = 7 \), then \( p(7) = 30 \cdot 7 = 210 \) minutes.
Therefore, the range of practice minutes \( p(n) \) as \( n \) varies from 0 to 7 is: \[ \text{Range: } {0, 30, 60, 90, 120, 150, 180, 210} \]
In interval notation, the range can be expressed as: \[ \text{Range: } [0, 210] \]
To summarize:
16) Domain: \([0, 7]\)
17) Range: \([0, 210]\)
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