A bacterial culture in a petri dish grows at an exponential rate. The petri dish has an area of 256 $$ \mathrm{~mm}^{2} $$, and the bacterial culture stops growing when it covers this area. The area in $$ \mathrm{mm}^{2} $$ that the bacteria cover each day is given by the function $$ f(x)=2^{x} $$. What is a reasonable domain for this function? A $$ 0

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To determine a reasonable domain for the function $$ f(x) = 2^x $$ representing the area covered by the bacteria each day, let's analyze the problem step by step.

1. **Understanding the Growth Function:**
   - The function $$ f(x) = 2^x $$ describes exponential growth, where $$ x $$ represents the number of days.
   - The bacteria start growing from an initial area (which could be considered as $$ f(0) = 1 $$ mm² if we assume they start from a single unit area).

2. **Determining When Growth Stops:**
   - The bacteria stop growing when they cover the entire petri dish, which has an area of 256 mm².
   - We need to find the value of $$ x $$ when $$ 2^x = 256 $$.

3. **Solving for $$ x $$:**
   $$
   2^x = 256
   $$
   $$
   256 = 2^8 \quad \text{(since $$ 2^8 = 256 $$)}
   $$
   $$
   \Rightarrow x = 8
   $$
   - This means that on the 8th day, the bacteria will have covered the entire petri dish.

4. **Determining the Reasonable Domain:**
   - The domain should start from day 0 (when growth begins) and end at day 8 (when growth stops).
   - Therefore, the reasonable domain is $$ 0 < x \leq 8 $$.

**Answer:**  
D) $$ 0 < x \leq 8 $$
The reasonable domain for the function is $$ 0 < x \leq 8 $$.

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