After taking a dose of medication, the amount of medicine remaining in a person's bloodstream, in milligrams, after xx hours can be modeled by the function f, of, x, equals, 110, left parenthesis, 0, point, 9, 1, right parenthesis, to the power x , .f(x)=110(0.91) 
x
 . Find and interpret the given function values and determine an appropriate domain for the function.
Round your answers to the nearest hundredth.
Answer
Attempt 1 out of 2

f, of, minus, 3, equalsf(−3)= , 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.
f, of, 36, equalsf(36)= , 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.
f, of, 10, point, 5, equalsf(10.5)= , 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.
Based on the observations above, it is clear that an appropriate domain for the function is .

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Let's break down the problem and compute the values you need. The function given is:

\[
f(x) = 110 \cdot (0.91)^x
\]

Where:
- \( f(x) \) represents the amount of medicine remaining in the bloodstream (in milligrams) after \( x \) hours.
- \( x \) represents the number of hours after taking the medication.

### 1. Evaluate \( f(-3) \)

\[
f(-3) = 110 \cdot (0.91)^{-3}
\]

Since \( x = -3 \) represents negative time, it's not realistic in the context of the problem. However, mathematically:

\[
f(-3) = 110 \cdot (0.91)^{-3} \approx 110 \cdot 1.3736 \approx 151.10
\]

Interpretation: \( f(-3) \) means 3 hours **before** taking the dose. But this is not valid since you can't have medicine in the bloodstream before it's taken.

### 2. Evaluate \( f(36) \)

\[
f(36) = 110 \cdot (0.91)^{36}
\]

\[
f(36) \approx 110 \cdot 0.0912 \approx 10.03
\]

Interpretation: After 36 hours, approximately 10.03 milligrams of medicine remain in the bloodstream.

### 3. Evaluate \( f(10.5) \)

\[
f(10.5) = 110 \cdot (0.91)^{10.5}
\]

\[
f(10.5) \approx 110 \cdot 0.3837 \approx 42.21
\]

Interpretation: After 10.5 hours, approximately 42.21 milligrams of medicine remain in the bloodstream.

### Domain of the function

Since \( x \) represents the number of hours after taking the dose, the domain must be restricted to non-negative values. Negative time values don't make sense in this context.

Thus, the appropriate domain is \( x \geq 0 \).

### Summary of Function Values:
- \( f(-3) = 151.10 \) milligrams (not valid in the context)
- \( f(36) = 10.03 \) milligrams (valid)
- \( f(10.5) = 42.21 \) milligrams (valid)

**Domain:** \( x \geq 0 \)

Would you like further details or have any other questions?

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Here are 5 related questions to expand on this topic:

1. What happens to the amount of medicine in the bloodstream as \( x \) increases indefinitely?
2. Can we use a different base for the exponential function to model a different medication?
3. How long would it take for the medicine to decrease to 1 milligram?
4. What is the half-life of the medication based on this function?
5. How would the function change if the medication decayed at a different rate?

**Tip:** Exponential decay models are often used in pharmacology to predict how substances diminish over time.

After taking a dose of medication, the amount of medicine remaining in a person's bloodstream, in milligrams, after xx hours can be modeled by the function f(x) = 110(0.91)^x. Find and interpret the given function values and determine an appropriate domain for the function. Round your answers to the nearest hundredth.

Learn how to analyze the exponential decay function f(x) = 110(0.91)^x to determine the amount of medication remaining in the bloodstream after a certain number of hours. This problem covers key concepts like exponential decay, interpreting function values, and determining the domain. Suitable for students in Grades 10-12.

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