Question

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(0.83)^{x}.F(x)=110(0.83) x . Find and interpret the given function values and determine an appropriate domain for the function.

Answer 1

Given:

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

$f(x)=110(0.83)^x$

To find:

Interpret the given function values and determine an appropriate domain for the function.

Solution:

The general form of an exponential function is

$f(x)=ab^x$

Where, a is the initial value, 0<b<1 is decay factor and b>1 is growth factor.

We have,

$f(x)=110(0.83)^x$

Here, 110 is the initial value and 0.83 is the decay factor.

It means, the amount of medicine in the person's bloodstream after taking the dose is 110 milligrams and the amount of medicine decreasing in the person's bloodstream with the decay factor 0.83 or decreasing at the rate of (1-0.83)=0.17=17%.

We know that an exponential function is defined for all real values of x but the time cannot be negative. So, x must be non negative.

$Domain:0\leq x$

$Domain:[0,\infty)$

We know that $(0.83)^x>0$ for any value of x. So, $110(0.83)^x>0$ for all values of x.

$Range:f(x)>0$

$Range:(0,\infty)$

Therefore, domain of the function is $[0,\infty)$ and the range is $(0,\infty)$.