Let's break down the problem and answer each part step by step.
1. What does [tex]\( t \)[/tex] represent?
In the context of the given problem, the variable [tex]\( t \)[/tex] represents the number of years since 2008. Thus, [tex]\( t = 0 \)[/tex] corresponds to the year 2008, [tex]\( t = 1 \)[/tex] corresponds to 2009, and so forth. This allows us to measure the population of bobcats in northern Arizona over time starting from the year 2008.
2. What is the domain for this function?
The domain of a function is the set of all possible input values (in this case, the values of [tex]\( t \)[/tex]). Since [tex]\( t \)[/tex] represents the number of years since 2008, we can denote [tex]\( t \)[/tex] as any non-negative number (i.e., [tex]\( t \)[/tex] cannot be negative because negative years wouldn't make sense in this context). Therefore, the domain for this function is [tex]\( t \geq 0 \)[/tex].
3. Which range values would not make sense for this function?
The range of a function refers to the set of all possible output values. Here, [tex]\( b(t) \)[/tex] represents the population of bobcats, which must be a non-negative integer. As a result, any negative numbers or non-integer values would not make sense for this function. Hence, in the context of modeling bobcat populations, only non-negative integers are meaningful, and values such as negative numbers or non-integers do not make sense.
4. Would the graph be continuous or discrete, and why?
The graph of the function [tex]\( b(t) = -0.32 t^2 + 2.7 t + 253 \)[/tex] would be continuous. This is because [tex]\( b(t) \)[/tex] is a quadratic equation, which describes a parabolic curve and is inherently a continuous function. This means that the function does not have any breaks, jumps, or gaps in its graph, and it smoothly progresses from one point to the next for all values in its domain. The continuity also aligns with the natural phenomenon of population over time, even though we may be interested in the population at discrete years.
To summarize:
- [tex]\( t \)[/tex] represents the number of years since 2008.
- The domain of this function is [tex]\( t \geq 0 \)[/tex].
- Range values that would not make sense include negative numbers and non-integers.
- The graph of this function would be continuous.
