Given the scenario, Anna wants to model the number of fruit flies over time as she prepares to bake banana bread. The function \( f(t) = 8(2)^t \) describes the number of fruit flies after \( t \) days, with the condition that she bakes the banana bread 5 days after shopping.
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Time Domain: Since Anna bakes the banana bread 5 days after shopping, the time \( t \) should be within the interval from 0 to 5 days. This provides us with the first part of the domain, which is:
\[ 0 \leq t \leq 5 \]
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Value Range for \( f \): To find the range of values for \( f(t) \) when \( t \) is from 0 to 5, we will compute \( f(0) \) and \( f(5) \):
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At \( t = 0 \): \[ f(0) = 8(2)^0 = 8 \cdot 1 = 8 \]
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At \( t = 5 \): \[ f(5) = 8(2)^5 = 8 \cdot 32 = 256 \]
Therefore, as \( t \) ranges from 0 to 5, \( f(t) \) will range from 8 to 256. Thus, we can summarize the range for \( f \):
\[ 8 \leq f \leq 256 \]
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Other Options: Analyzing the provided options:
- \( 0 \leq f \leq 256 \): This is partially correct because \( f \) starts at 8, but the range from zero to 256 does not reflect the actual values \( f(t) \) will take.
- \( 0 \leq f \leq 5 \): This is incorrect because \( f(t) \) starts at 8 and goes up to 256.
- \( 0 \leq t \leq 256 \): This is also incorrect since \( t \) can only go up to 5.
Based on this analysis, the appropriate domain for the problem is:
\[ 0 \leq t \leq 5 \]
And the range of \( f \) is:
\[ 8 \leq f \leq 256 \]