Question

Elena plays the piano for 30 minutes each practice day. The total number of minutes pthat Elena practiced last week is a function of n, the number of practice days.Find the domain and range for this function.16) Domain: 17)Range:

Answer 1

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 \( n \):

1. **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:
\[
\text{Domain: } [0, 7]
\]

2. **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: \([0, 210]\)