Question

question
after taking a dose of medication, the amount of medicine remaining in a person’s 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.
answer attempt 1 out of 2
( f(-6) = ) ( 292.67 ), meaning ( -6 ) hours after taking the dose, there are ( 292.67 ) milligrams of medicine remaining in the person’s bloodstream. this interpretation does not make sense in the context of the problem.
( f(6) = 42.92 ), meaning ( 6 ) hours after taking the dose, there are ( 42.92 ) milligrams of medicine remaining in the person’s bloodstream. this interpretation makes sense
( f(7.5) = 35.54 ), meaning ( 7.5 ) hours after taking the dose, there are ( 35.54 ) milligrams of medicine remaining in the person’s bloodstream. this interpretation makes sense
based on the observations above, it is clear that an appropriate domain for the function is integers in ( a leq x leq b ) for ( square leq x leq square )

Explanation:

Step1: Define the given function

The function is $f(x) = 95(0.84)^x$, where $x$ = hours after dose, $f(x)$ = mg of medicine.

Step2: Calculate $f(-6)$

Substitute $x=-6$ into the function.
$f(-6) = 95(0.84)^{-6} = 95 \times \frac{1}{(0.84)^6}$
First calculate $(0.84)^6 \approx 0.351298$, then $\frac{1}{0.351298} \approx 2.8466$, so $95 \times 2.8466 \approx 270.43$ (rounded to nearest hundredth).

Step3: Calculate $f(6)$

Substitute $x=6$ into the function.
$f(6) = 95(0.84)^6 \approx 95 \times 0.351298 \approx 33.37$

Step4: Calculate $f(7.5)$

Substitute $x=7.5$ into the function.
$f(7.5) = 95(0.84)^{7.5}$
First calculate $(0.84)^{7.5} = (0.84)^6 \times (0.84)^{1.5} \approx 0.351298 \times 0.783 \approx 0.2751$, then $95 \times 0.2751 \approx 26.13$

Step5: Determine valid domain

Time cannot be negative, and when $x$ increases, $f(x)$ approaches 0 (medicine is eliminated). A practical domain starts at $x=0$ (time of dose) and can go to a point where the medicine is negligible, but minimally $x \geq 0$. For integer values, $0 \leq x \leq n$ where $n$ is a large enough integer, but the core valid lower bound is 0.

Answer:

• $f(-6) \approx 270.43$, meaning -6 hours after taking the dose, there are 270.43 milligrams of medicine remaining in the person's bloodstream. This interpretation does NOT make sense in the context of the problem.
• $f(6) \approx 33.37$, meaning 6 hours after taking the dose, there are 33.37 milligrams of medicine remaining in the person's bloodstream. This interpretation makes sense in the context of the problem.
• $f(7.5) \approx 26.13$, meaning 7.5 hours after taking the dose, there are 26.13 milligrams of medicine remaining in the person's bloodstream. This interpretation makes sense in the context of the problem.
• An appropriate domain for the function (integer values) is $\boldsymbol{0 \leq x \leq b}$ where $b$ is a positive integer (e.g., 24, when the medicine is nearly eliminated). The minimum valid integer value is 0.