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 __________.

Answer 1

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 \[x \in [-6, \infty)\].