Question

iodine iodine 131 is a radioisotope that is related to nuclear energy, medical diagnostic and treatment procedures, and natural gas production. a scientist is testing 50 milligrams of iodine 131. the scientist knows that the half-life of iodine 131 is about 8.02 days. the function $y = 50(0.92)^x$ represents the amount of iodine 131 remaining in milligrams $y$ after $x$ days.
a. determine whether the function represents exponential growth or exponential decay. the function represents select choice
b. identify the relevant domain and range.
because time select choice, the relevant domain is select choice.
because the amount of iodine 131 select choice, and the amount when $x = 0$ is select choice mg, the relevant range is
select choice
$\\{y \mid y > 0\\}$
$\\{y \mid 0 < y \leq 25\\}$
$\\{y \mid 0 < y \leq 50\\}$
$\\{y \mid y \leq 50\\}$

Explanation:

Step1: Classify exponential function

An exponential function has the form $y = a(b)^x$. If $0 < b < 1$, it represents decay; if $b > 1$, it represents growth. Here, $b = 0.92$, which is between 0 and 1.

Step2: Identify domain (time)

Time $x$ cannot be negative, so the domain is all non-negative real numbers: $\{x \mid x \geq 0\}$.

Step3: Find initial amount

When $x=0$, substitute into the function: $y = 50(0.92)^0 = 50(1) = 50$ mg.

Step4: Identify range (amount of iodine)

The amount of iodine decreases over time but can never be negative, starting from 50 mg. So the range is $\{y \mid 0 < y \leq 50\}$.

Answer:

a. The function represents exponential decay
b. Because time cannot be negative, the relevant domain is $\{x \mid x \geq 0\}$.
Because the amount of iodine 131 starts at 50 mg, the amount when $x=0$ is 50 mg, the relevant range is $\{y \mid 0 < y \leq 50\}$