To determine a reasonable domain for the function [tex]\( f(x) = 2^x \)[/tex] given that the total area covered by the bacteria is limited to [tex]\( 256 \, \text{mm}^2 \)[/tex], follow these steps:
1. Recognize that the function [tex]\( f(x) = 2^x \)[/tex] represents exponential growth. This means the area covered by the bacteria increases exponentially as [tex]\( x \)[/tex], the number of days, increases.
2. The bacterial culture stops growing when it covers a total area of [tex]\( 256 \, \text{mm}^2 \)[/tex]. Therefore, we need to determine the value of [tex]\( x \)[/tex] such that [tex]\( 2^x \)[/tex] does not exceed [tex]\( 256 \)[/tex].
3. To find the maximum value of [tex]\( x \)[/tex] that satisfies the condition [tex]\( 2^x \leq 256 \)[/tex], we observe that:
[tex]\[
2^8 = 256
\][/tex]
This implies that [tex]\( x \)[/tex] can be up to 8 days long where the function [tex]\( f(x) \)[/tex] reaches the maximum allowable area of [tex]\( 256 \, \text{mm}^2 \)[/tex].
4. Therefore, a reasonable domain for the function must ensure that the bacterial growth stays within the range and does not exceed the petri dish area. This domain can be expressed as:
[tex]\[
0 < x \leq 8
\][/tex]
Given the answer choices, the correct domain that ensures the bacterial culture covers an area within [tex]\( 256 \, \text{mm}^2 \)[/tex] is:
D [tex]\( 0 < x \leq 8 \)[/tex]
