Answer :
Let's analyze the given function [tex]\( f(x) = 110 (0.91)^x \)[/tex] and its values at specific points.
1. Find and interpret [tex]\( f(-3) \)[/tex]:
- Upon calculating [tex]\( f(-3) \)[/tex], we get approximately 145.97 milligrams of medicine.
- The value [tex]\( f(-3) = 145.97 \)[/tex] does not make sense in the context of this problem, as it suggests an amount of medicine 3 hours before the dose was taken, which is not applicable.
So, the answer is:
[tex]\( f(-3) = 145.97 \)[/tex], meaning 3 hours after taking the dose, there are 145.97 milligrams of medicine remaining in the person's bloodstream. This interpretation does not make sense in the context of the problem.
2. Find and interpret [tex]\( f(36) \)[/tex]:
- Upon calculating [tex]\( f(36) \)[/tex], we get approximately 3.69 milligrams of medicine.
- This means 36 hours after taking the dose, there are 3.69 milligrams of medicine remaining in the person's bloodstream.
So, the answer is:
[tex]\( f(36) = 3.69 \)[/tex], meaning 36 hours after taking the dose, there are 3.69 milligrams of medicine remaining in the person's bloodstream. This interpretation makes sense in the context of the problem.
3. Find and interpret [tex]\( f(10.5) \)[/tex]:
- Upon calculating [tex]\( f(10.5) \)[/tex], we get approximately 40.86 milligrams of medicine.
- This means 10.5 hours after taking the dose, there are 40.86 milligrams of medicine remaining in the person's bloodstream.
So, the answer is:
[tex]\( f(10.5) = 40.86 \)[/tex], meaning 10.5 hours after taking the dose, there are 40.86 milligrams of medicine remaining in the person's bloodstream. This interpretation makes sense in the context of the problem.
4. Determine an appropriate domain for the function [tex]\( f(x) \)[/tex]:
- Since time cannot be negative, the domain of the function [tex]\( f(x) = 110 (0.91)^x \)[/tex] is [tex]\( [0, ∞) \)[/tex].
So, the appropriate domain for the function is:
The domain is [tex]\( [0, ∞) \)[/tex].
Summarizing all the answers:
- [tex]\( f(-3) = 145.97 \)[/tex], meaning 3 hours before taking the dose, there are 145.97 milligrams of medicine remaining in the person's bloodstream. This interpretation does not make sense in the context of the problem.
- [tex]\( f(36) = 3.69 \)[/tex], meaning 36 hours after taking the dose, there are 3.69 milligrams of medicine remaining in the person's bloodstream.
- [tex]\( f(10.5) = 40.86 \)[/tex], meaning 10.5 hours after taking the dose, there are 40.86 milligrams of medicine remaining in the person's bloodstream.
- The domain is [tex]\( [0, ∞) \)[/tex].
1. Find and interpret [tex]\( f(-3) \)[/tex]:
- Upon calculating [tex]\( f(-3) \)[/tex], we get approximately 145.97 milligrams of medicine.
- The value [tex]\( f(-3) = 145.97 \)[/tex] does not make sense in the context of this problem, as it suggests an amount of medicine 3 hours before the dose was taken, which is not applicable.
So, the answer is:
[tex]\( f(-3) = 145.97 \)[/tex], meaning 3 hours after taking the dose, there are 145.97 milligrams of medicine remaining in the person's bloodstream. This interpretation does not make sense in the context of the problem.
2. Find and interpret [tex]\( f(36) \)[/tex]:
- Upon calculating [tex]\( f(36) \)[/tex], we get approximately 3.69 milligrams of medicine.
- This means 36 hours after taking the dose, there are 3.69 milligrams of medicine remaining in the person's bloodstream.
So, the answer is:
[tex]\( f(36) = 3.69 \)[/tex], meaning 36 hours after taking the dose, there are 3.69 milligrams of medicine remaining in the person's bloodstream. This interpretation makes sense in the context of the problem.
3. Find and interpret [tex]\( f(10.5) \)[/tex]:
- Upon calculating [tex]\( f(10.5) \)[/tex], we get approximately 40.86 milligrams of medicine.
- This means 10.5 hours after taking the dose, there are 40.86 milligrams of medicine remaining in the person's bloodstream.
So, the answer is:
[tex]\( f(10.5) = 40.86 \)[/tex], meaning 10.5 hours after taking the dose, there are 40.86 milligrams of medicine remaining in the person's bloodstream. This interpretation makes sense in the context of the problem.
4. Determine an appropriate domain for the function [tex]\( f(x) \)[/tex]:
- Since time cannot be negative, the domain of the function [tex]\( f(x) = 110 (0.91)^x \)[/tex] is [tex]\( [0, ∞) \)[/tex].
So, the appropriate domain for the function is:
The domain is [tex]\( [0, ∞) \)[/tex].
Summarizing all the answers:
- [tex]\( f(-3) = 145.97 \)[/tex], meaning 3 hours before taking the dose, there are 145.97 milligrams of medicine remaining in the person's bloodstream. This interpretation does not make sense in the context of the problem.
- [tex]\( f(36) = 3.69 \)[/tex], meaning 36 hours after taking the dose, there are 3.69 milligrams of medicine remaining in the person's bloodstream.
- [tex]\( f(10.5) = 40.86 \)[/tex], meaning 10.5 hours after taking the dose, there are 40.86 milligrams of medicine remaining in the person's bloodstream.
- The domain is [tex]\( [0, ∞) \)[/tex].
