Question
after taking a dose of medication, the amount of medicine remaining in a persons bloodstream, in milligrams, after x hours can be modeled by the function f(x)=95(0.84)^X. Find and interpret the given function values and determine an appropriate domain for the function. Round your answers to the nearest hundredth. 1. f(-6)=_______, meaning ______hours after taking the dose, there are _______milligrams of medicine remaining in the person's bloodstream. The interpretation _________in the context of the problem. 2. f(6)=_______, meaning _______hours after taking the dose, there are _________milligrams of medicine remaining in the person's bloodstream. This interpretation _______in the context of the problem. 3. f(7.5)=______, meaning ________hours after taking the dose, there are _______milligrams of medicine remaining in person's bloodstream. This interpretation _________in the context of the problem. Based on the observations above, it is clear that an appropriate domain for the function is __________.
Answers
GPT-4o mini
To evaluate the function \( f(x) = 95(0.84)^x \) for different values of \( x \) and interpret the results, we will calculate \( f(-6) \), \( f(6) \), and \( f(7.5) \).
### 1. Calculating \( f(-6) \)
\[
f(-6) = 95(0.84)^{-6}
\]
\[
= 95 \times \frac{1}{(0.84)^6} \approx 95 \times 1.50 \approx 142.5
\]
Thus,
\[
f(-6) = 142.5
\]
- Meaning: **6 hours before taking the dose**, there are approximately **142.5 milligrams** of medicine remaining in the person's bloodstream.
- The interpretation is **relevant since it suggests the amount present before the medication was taken**.
### 2. Calculating \( f(6) \)
\[
f(6) = 95(0.84)^6
\]
\[
\approx 95 \times 0.3349 \approx 31.7
\]
Thus,
\[
f(6) \approx 31.70
\]
- Meaning: **6 hours after taking the dose**, there are approximately **31.70 milligrams** of medicine remaining in the person's bloodstream.
- This interpretation **indicates the declining amount of medicine over time**.
### 3. Calculating \( f(7.5) \)
\[
f(7.5) = 95(0.84)^{7.5}
\]
\[
\approx 95 \times 0.2711 \approx 25.7
\]
Thus,
\[
f(7.5) \approx 25.67
\]
- Meaning: **7.5 hours after taking the dose**, there are approximately **25.67 milligrams** of medicine remaining in the person's bloodstream.
- This interpretation **continues to show the decrease in the amount of medicine over time**.
### Appropriate Domain for the Function
The domain of the function is the set of real numbers, but in this context, \( x \) should represent time in hours around the administration of the medicine. Therefore, considering the context of the problem, an appropriate domain is:
\[
x \geq -6 \text{ (to allow for prior time relative to the dose)}
\]
So, the overall domain can be expressed as:
\[
\text{Domain}: x \in [-6, \infty)
\]
### Summary of Answers
1. \( f(-6) \approx 142.5 \), meaning **6 hours before taking the dose**, there are **142.5 milligrams** of medicine remaining in the person's bloodstream. The interpretation **is relevant since it suggests the amount present before the medication was taken**.
2. \( f(6) \approx 31.70 \), meaning **6 hours after taking the dose**, there are **31.70 milligrams** of medicine remaining in the person's bloodstream. This interpretation **indicates the declining amount of medicine over time**.
3. \( f(7.5) \approx 25.67 \), meaning **7.5 hours after taking the dose**, there are **25.67 milligrams** of medicine remaining in the person's bloodstream. This interpretation **continues to show the decrease in the amount of medicine over time**.
4. An appropriate domain for the function is **\[x \in [-6, \infty)\]**.
### 1. Calculating \( f(-6) \)
\[
f(-6) = 95(0.84)^{-6}
\]
\[
= 95 \times \frac{1}{(0.84)^6} \approx 95 \times 1.50 \approx 142.5
\]
Thus,
\[
f(-6) = 142.5
\]
- Meaning: **6 hours before taking the dose**, there are approximately **142.5 milligrams** of medicine remaining in the person's bloodstream.
- The interpretation is **relevant since it suggests the amount present before the medication was taken**.
### 2. Calculating \( f(6) \)
\[
f(6) = 95(0.84)^6
\]
\[
\approx 95 \times 0.3349 \approx 31.7
\]
Thus,
\[
f(6) \approx 31.70
\]
- Meaning: **6 hours after taking the dose**, there are approximately **31.70 milligrams** of medicine remaining in the person's bloodstream.
- This interpretation **indicates the declining amount of medicine over time**.
### 3. Calculating \( f(7.5) \)
\[
f(7.5) = 95(0.84)^{7.5}
\]
\[
\approx 95 \times 0.2711 \approx 25.7
\]
Thus,
\[
f(7.5) \approx 25.67
\]
- Meaning: **7.5 hours after taking the dose**, there are approximately **25.67 milligrams** of medicine remaining in the person's bloodstream.
- This interpretation **continues to show the decrease in the amount of medicine over time**.
### Appropriate Domain for the Function
The domain of the function is the set of real numbers, but in this context, \( x \) should represent time in hours around the administration of the medicine. Therefore, considering the context of the problem, an appropriate domain is:
\[
x \geq -6 \text{ (to allow for prior time relative to the dose)}
\]
So, the overall domain can be expressed as:
\[
\text{Domain}: x \in [-6, \infty)
\]
### Summary of Answers
1. \( f(-6) \approx 142.5 \), meaning **6 hours before taking the dose**, there are **142.5 milligrams** of medicine remaining in the person's bloodstream. The interpretation **is relevant since it suggests the amount present before the medication was taken**.
2. \( f(6) \approx 31.70 \), meaning **6 hours after taking the dose**, there are **31.70 milligrams** of medicine remaining in the person's bloodstream. This interpretation **indicates the declining amount of medicine over time**.
3. \( f(7.5) \approx 25.67 \), meaning **7.5 hours after taking the dose**, there are **25.67 milligrams** of medicine remaining in the person's bloodstream. This interpretation **continues to show the decrease in the amount of medicine over time**.
4. An appropriate domain for the function is **\[x \in [-6, \infty)\]**.