Question

A bacterial culture in a petri dish grows at an exponential rate. The petri dish has an area of 256mm², and the bacterial culture stops growing when it covers this area. The area in mm² that the bacteria cover each day is given by the function f(x) = 2x. What is a reasonable domain for this function?

1) All real numbers
2) Positive integers
3) Positive real numbers
4) Non-negative integers

Answer 1

The most reasonable domain for the function representing the exponential growth of bacteria is the set of non-negative integers, as it aligns with the concept of daily growth measurement and the physical limits imposed by the petri dish area.

Step-by-step explanation:

The bacterial culture in question exhibits exponential growth, as represented by the function f(x) = 2x where x is the number of days. For a function that models natural occurrences such as bacterial growth, the domain refers to the set of all possible input values (in this context, time in days) that the function can accept meaningfully. The culture grows each day and presumably starts at day 0; thus, it does not make sense to consider negative time values.

A reasonable domain for this function would be the set of non-negative integers (0, 1, 2, 3, ...), as the bacteria grow quantized by days, and we cannot have a fraction of a day. Moreover, the question specifies that the petri dish area is 256 mm2 and the bacterial culture stops growing when it covers this area, which suggests a finite end to the growth period once the area is completely covered. Therefore, the domain does not need to extend infinitely but is rather limited to the number of days it takes for the bacteria to cover the petri dish.

Considering all this, the most reasonable domain for the function f(x) given the scenario is option (4) Non-negative integers.