To analyze the exponential function [tex]\( y = 2^{-x} \)[/tex], we'll need to determine whether it represents growth or decay and then calculate the percentage rate of increase or decrease per unit of [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Identify whether the function represents growth or decay:
- The function is [tex]\( y = 2^{-x} \)[/tex]. This can also be written as [tex]\( y = \left(\frac{1}{2}\right)^x \)[/tex].
- Since [tex]\( \frac{1}{2} \)[/tex] (which is less than 1) is raised to the power of [tex]\( x \)[/tex], this represents an exponential decay function. Therefore, the function [tex]\( y = 2^{-x} \)[/tex] represents decay.
2. Determine the percentage rate of decrease per unit of [tex]\( x \)[/tex]:
- In an exponential function of the form [tex]\( y = a^x \)[/tex], where [tex]\( 0 < a < 1 \)[/tex], the base [tex]\( a = \frac{1}{2} \)[/tex] indicates decay.
- The general form for the percentage rate of decrease per unit [tex]\( x \)[/tex] is derived by expressing the base [tex]\( a \)[/tex] as [tex]\( a = 1 - r \)[/tex], where [tex]\( r \)[/tex] is the decay rate.
3. Convert the base to find the decay rate:
- Here, the base [tex]\( a = \frac{1}{2} \)[/tex]:
[tex]\[
\frac{1}{2} = 1 - r
\][/tex]
- Solving for [tex]\( r \)[/tex]:
[tex]\[
r = 1 - \frac{1}{2} = \frac{1}{2}
\][/tex]
4. Convert the decay rate into a percentage:
- Multiply [tex]\( r \)[/tex] by 100 to convert it to a percentage:
[tex]\[
\text{Percentage rate of decrease} = \frac{1}{2} \times 100\% = 50\%
\][/tex]
Therefore, the function [tex]\( y = 2^{-x} \)[/tex] represents a decay, and the percentage rate of decrease per unit of [tex]\( x \)[/tex] is [tex]\( 50 \%\)[/tex].
