To fully understand Allen's function, let's break down the components of the given expression [tex]\( f(x) = 5\left(\frac{1}{10}\right)^x \)[/tex].
1. Initial Value at [tex]\( x=0 \)[/tex]:
At [tex]\( x=0 \)[/tex]:
[tex]\[
f(0) = 5 \left(\frac{1}{10}\right)^0 = 5 \times 1 = 5
\][/tex]
This confirms that the initial value of the function, or the y-intercept, is indeed [tex]\( 5 \)[/tex].
2. Rate of Decay:
The expression given is [tex]\( \left(\frac{1}{10}\right)^x \)[/tex], which indicates an exponential decay because the base [tex]\(\frac{1}{10}\)[/tex] is less than 1.
The term [tex]\(\frac{1}{10}\)[/tex] can also be represented as [tex]\( 0.1 \)[/tex]. This highlights that the rate of decay is [tex]\( 0.1 \)[/tex] or 10%.
3. Exponential Function Form:
An exponential decay function can generally be expressed as:
[tex]\[
f(x) = \text{initial value} \times (\text{decay rate})^x
\][/tex]
Here, the initial value is [tex]\( 5 \)[/tex], and the decay rate is [tex]\( \frac{1}{10} \)[/tex] (or 0.1).
Completing the statement:
Allen's function is an exponential decay function because the exponential function [tex]\( f \)[/tex] should have a base equal to the expression [tex]\( 0.1 \)[/tex] (or [tex]\(\frac{1}{10}\)[/tex]) and an initial value of [tex]\( 5 \)[/tex].
So, the completed statement is:
Allen's function is an exponential decay function because the exponential function [tex]\( f \)[/tex] should have a base equal to the expression [tex]\( 0.1 \)[/tex] and an initial value of [tex]\( 5 \)[/tex].
