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Bethany is a biologist who has been measuring and recording the rabbit population at a park over the last several years. She has noticed that a trigonometric function can model the population, which reaches a maximum population, [tex]m[/tex], each summer and a minimum population, [tex]n[/tex], each winter. If she uses [tex]p(t)[/tex] to represent the rabbit population [tex]t[/tex] years from the start of her study, what is the domain of function [tex]p[/tex]?

A. [tex](-\infty, \infty)[/tex]
B. [tex][0, \infty)[/tex]
C. [tex][0, m][/tex]
D. [tex][m, n][/tex]

Sagot :

GinnyAnsweridn
Sure, let's analyze the situation step by step to determine the domain of the function [tex]\( p \)[/tex] that models the rabbit population [tex]\( f \)[/tex] years from the start of Bethany's study.

1. Understanding the context:
- We're dealing with a function [tex]\( p(t) \)[/tex] that represents the rabbit population over time.
- Key information provided: The population fluctuates between a maximum [tex]\( m \)[/tex] in the summer and a minimum [tex]\( n \)[/tex] in the winter.

2. Defining Domain:
- The domain of a function is the set of all possible input values (in this case, time in years, [tex]\( t \)[/tex]).
- Since [tex]\( t \)[/tex] represents time measured in years from the start of Bethany's study, we need to consider realistic values for [tex]\( t \)[/tex].

3. Analyzing options:
- Option A: [tex]\((- \infty, \infty)\)[/tex]
- This option suggests that time could be any real number, both negative and positive.
- However, time [tex]\( f \)[/tex] years from the start of the study can only be non-negative because we don't measure time in negative years.

- Option B: [tex]\([0, \infty)\)[/tex]
- This option suggests that time starts from 0 years (the start of the study) and extends forward indefinitely.
- Since we are measuring years from the start of the study, this makes sense because it aligns with real-world scenarios.

- Option C: [tex]\([0, m]\)[/tex]
- This option suggests that time ranges from 0 to [tex]\( m \)[/tex].
- This does not make sense because [tex]\( m \)[/tex] is the maximum population, not a time measurement.

- Option D: [tex]\([m, n]\)[/tex]
- This option also does not make sense for representing time because [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are values of the population, not time.

Therefore, after analyzing all the options, the correct domain that reasonably fits the situation for the rabbit population [tex]\( p(t) \)[/tex] function is:

[tex]\[ \boxed{[0, \infty)} \][/tex]

This is option B.
Time cannot be negative and can extend indefinitely into the future, hence the domain starts from 0 and goes up to positive infinity.
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