Let's break this problem down step by step.

The given function for the amount of medication remaining in a person's bloodstream is:

$f(x) = 60(0.85)^x$

Where:

• $f(x)$ represents the amount of medicine in milligrams.
• $x$ represents the number of hours after taking the dose.
• The initial amount of medicine is 60 milligrams.

Step 1: Find $f(-3)$

We are asked to evaluate $f(-3)$, which means we need to plug $x = -3$ into the function:

$f(-3) = 60(0.85)^{-3}$

We calculate the value of $(0.85)^{-3}$ first:

$0.85^{-3} = \frac{1}{(0.85)^3} \approx \frac{1}{0.614125} \approx 1.629$

Now calculate $f(-3)$:

$f(-3) = 60 \times 1.629 \approx 97.74$

So, $f(-3) \approx 97.74$ milligrams.

Interpretation of $f(-3)$:

This interpretation does NOT make sense in the context of the problem, because negative hours don't have any physical meaning here. Hours after taking the medication should be non-negative (i.e., after 0 hours, not before).

Step 2: Find $f(24)$

Next, let's find $f(24)$, which represents the amount of medicine after 24 hours:

$f(24) = 60(0.85)^{24}$

First, calculate $(0.85)^{24}$:

$0.85^{24} \approx 0.035$

Now calculate $f(24)$:

$f(24) = 60 \times 0.035 \approx 2.10$

So, $f(24) \approx 2.10$ milligrams.

Interpretation of $f(24)$:

This interpretation makes sense because after 24 hours, there are 2.10 milligrams of medicine remaining in the bloodstream, which is a reasonable amount.

Step 3: Find $f(6.5)$

Now, we evaluate $f(6.5)$, which represents the amount of medicine remaining after 6.5 hours:

$f(6.5) = 60(0.85)^{6.5}$

First, calculate $(0.85)^{6.5}$:

$0.85^{6.5} \approx 0.335$

Now calculate $f(6.5)$:

$f(6.5) = 60 \times 0.335 \approx 20.10$

So, $f(6.5) \approx 20.10$ milligrams.

Interpretation of $f(6.5)$:

This interpretation makes sense because after 6.5 hours, there are 20.10 milligrams of medicine remaining in the bloodstream, which is a reasonable amount.

Step 4: Determine an appropriate domain for the function

The domain of the function refers to the possible values of $x$ (the number of hours). Since $x$ represents the time after taking the medication, it must be a non-negative number. Therefore, the domain of the function is:

$x \geq 0$

Thus, the appropriate domain for the function is $[0, \infty)$.

Final Answer:

• $f(-3) = 97.74$ milligrams, which does NOT make sense.
• $f(24) = 2.10$ milligrams, which makes sense.
• $f(6.5) = 20.10$ milligrams, which makes sense.

The appropriate domain for the function is $[0, \infty)$, meaning the function applies for $x \geq 0$ hours.

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Would you like a more detailed explanation or have any specific questions about the problem?

Here are 5 related questions to explore further:

1. How can the rate of decay of the medicine be represented mathematically?
2. How would the graph of the function look, and what are its key features?
3. What would happen if the factor $0.85$ were changed to a different number, like $0.9$?
4. How can we calculate the time it takes for the medicine to reach a certain amount (e.g., 5 milligrams)?
5. Why does the function model exponential decay in this situation?

Tip: The base of an exponential function (here $0.85$) represents the rate at which the quantity decreases. Values between 0 and 1 indicate decay, while values greater than 1 indicate growth.