Let's solve the problem step by step:

The function given is:

$f(x) = 75(0.84)^x$

where $f(x)$ represents the amount of medicine remaining in milligrams after $x$ hours. We need to calculate specific values and interpret the results, as well as determine an appropriate domain for the function.

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Step 1: Calculate $f(-4)$

$f(-4) = 75(0.84)^{-4}$ Using the property of negative exponents, $(a^{-b} = 1/a^b)$: $f(-4) = 75 \cdot \frac{1}{(0.84)^4} \approx 75 \cdot 1.5916 = 119.37$ Interpretation: $f(-4) = 119.37$ means 4 hours before the medication was taken, there would be 119.37 mg of medicine in the bloodstream. This interpretation does not make sense in the context of the problem, as the function is only valid after the dose is taken.

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Step 2: Calculate $f(42)$

$f(42) = 75(0.84)^{42}$ $f(42) \approx 75 \cdot (0.000037) \approx 0.0028$ Interpretation: $f(42) = 0.0028$ means 42 hours after taking the medication, approximately 0.0028 mg of medicine remains in the bloodstream. This interpretation is valid in the context of the problem, but the amount of medicine is effectively negligible.

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Step 3: Calculate $f(10.5)$

$f(10.5) = 75(0.84)^{10.5}$ $f(10.5) \approx 75 \cdot 0.2351 \approx 17.63$ Interpretation: $f(10.5) = 17.63$ means 10.5 hours after taking the medication, approximately 17.63 mg of medicine remains in the bloodstream. This interpretation makes sense and is consistent with the function.

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Step 4: Determine the Domain

The domain of the function should correspond to meaningful $x$-values in the context of the problem. Since $x$ represents the number of hours after taking the dose, the domain must start at $x = 0$ (when the dose is taken) and extend indefinitely into the future. However, practically, the function is only meaningful as long as $f(x) > 0$, which happens for $x \geq 0$.

Thus, the appropriate domain is:

$x \geq 0$

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Final Answer

1. $f(-4) = 119.37$: This interpretation does not make sense in the context of the problem.
2. $f(42) = 0.0028$: This interpretation makes sense, but the medicine is effectively negligible at this point.
3. $f(10.5) = 17.63$: This interpretation makes sense and is realistic.
4. Domain: $x \geq 0$.

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Follow-Up Questions

1. What would $f(20)$ represent in the context of the problem?
2. How does the value of $f(x)$ change as $x$ increases?
3. What is the half-life of the medicine based on this function?
4. Why does the domain exclude negative values of $x$?
5. How could this function be used to estimate when the medication becomes ineffective?

Tip: When analyzing exponential decay, always consider the initial value and the base to determine how quickly the quantity decreases.