In the given function \( f(x) = 64(0.50)^x \), \( f(x) \) represents the number of teams remaining after \( x \) rounds of a tournament where teams are eliminated in pairs (i.e., only the winning teams progress to the next round).
To define an appropriate range for the situation:
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Initial Teams: At the start of the tournament (when \( x = 0 \)), \( f(0) = 64(0.50)^0 = 64 \) teams play.
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Number of Teams: As each round progresses, the number of teams halves:
- After 1 round (\( x = 1 \)): \( f(1) = 64(0.50)^1 = 32 \)
- After 2 rounds (\( x = 2 \)): \( f(2) = 64(0.50)^2 = 16 \)
- After 3 rounds (\( x = 3 \)): \( f(3) = 64(0.50)^3 = 8 \)
- After 4 rounds (\( x = 4 \)): \( f(4) = 64(0.50)^4 = 4 \)
- After 5 rounds (\( x = 5 \)): \( f(5) = 64(0.50)^5 = 2 \)
- After 6 rounds (\( x = 6 \)): \( f(6) = 64(0.50)^6 = 1 \)
- After 7 rounds (\( x = 7 \)): \( f(7) = 64(0.50)^7 = 0.5 \)
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Termination of the Tournament: The tournament logically ends when there is at least 1 team left. We cannot have a fraction of a team, so \( f(x) \) must be an integer.
Thus, from the calculations above, we see that:
- The maximum number of teams is 64 (at \( x = 0 \)).
- The minimum number of teams that can be considered to still be participating in the tournament is 1 (at \( x = 6 \)).
- Beyond \( x = 6 \), \( f(x) \) produces fractional values, indicating that a team cannot be halved further in the context of this tournament setup.
Therefore, the appropriate range for \( f(x) \) is the set of integers from 1 to 64.
In interval notation, this can be written as: \[ \text{Range} = { 64, 32, 16, 8, 4, 2, 1 } \] or in terms of integers: \[ \text{Range} = { n \in \mathbb{Z} \mid 1 \leq n \leq 64 } \]
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