A bacteria is growing by a factor of 2 every hour from 1 p.m. to 11 p.m. The function below shows the number of bacterial cells, f(x), after x hours from 1 p.m.:
[tex]$f(x)=20(2)^x$[/tex]
Which of the following is a reasonable domain for the function?
A. All positive integers
B. [tex]$1 \leq x \leq 11$[/tex]
C. [tex]$0 \leq x \leq 20$[/tex]
D. [tex]$0 \leq x \leq 10$[/tex]
Answer (1)
Determine the time interval: The bacteria grows from 1 p.m. to 11 p.m., which is a total of 10 hours.
Define the variable: x represents the number of hours after 1 p.m.
Establish the domain: The domain starts at 0 (1 p.m.) and ends at 10 (11 p.m.).
State the final domain: The reasonable domain for the function is 0 ≤ x ≤ 10 .
Explanation
Understanding the Problem We are given a function that models the growth of bacteria over time, specifically from 1 p.m. to 11 p.m. The function is f ( x ) = 20 ( 2 ) x , where x represents the number of hours after 1 p.m. Our goal is to determine a reasonable domain for this function, considering the time interval during which the bacteria is growing.
Calculating the Time Interval The bacteria grows from 1 p.m. to 11 p.m. To find the number of hours in this interval, we subtract the starting time from the ending time: 11 p.m. - 1 p.m. = 10 hours. Since x represents the number of hours after 1 p.m., the domain should start at 0 (representing 1 p.m.) and end at 10 (representing 11 p.m.).
Determining the Domain Therefore, a reasonable domain for the function is 0 { l e q } x { l e q } 10 , which means x can take any value between 0 and 10, inclusive.
Examples
Understanding the domain of a function is crucial in many real-world applications. For instance, if you're tracking the spread of a virus, the domain would represent the time period over which you're observing the infection. Similarly, in finance, if you're modeling the growth of an investment, the domain would be the investment period. In this bacteria growth example, knowing the domain helps us understand the time frame over which the bacterial population is accurately represented by the given function. This is important for making predictions and informed decisions based on the model.
