To determine the multiplicative rate of change for the exponential function [tex]\( f(x) = 2 \left( \frac{5}{2} \right)^{-x} \)[/tex], follow these steps:
1. Rewrite the given function:
The given function is [tex]\( f(x) = 2 \left( \frac{5}{2} \right)^{-x} \)[/tex].
2. Simplify the exponent:
Recall that raising a fraction to a negative exponent inverts the fraction. Thus, we can rewrite the function as:
[tex]\[
f(x) = 2 \left( \frac{2}{5} \right)^x
\][/tex]
3. Identify the base:
The standard form of an exponential function is [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base. From our rewritten function, [tex]\( f(x) = 2 \left( \frac{2}{5} \right)^x \)[/tex], we see that the base [tex]\( b \)[/tex] is [tex]\( \frac{2}{5} \)[/tex].
4. Determine the multiplicative rate of change:
The multiplicative rate of change for the exponential function is the base, [tex]\( \frac{2}{5} \)[/tex].
5. Convert the fraction to a decimal:
[tex]\( \frac{2}{5} \)[/tex] can be expressed as a decimal:
[tex]\[
\frac{2}{5} = 0.4
\][/tex]
So, the multiplicative rate of change for the exponential function [tex]\( f(x) = 2 \left( \frac{5}{2} \right)^{-x} \)[/tex] is [tex]\( 0.4 \)[/tex].
The correct answer is:
[tex]\[
\boxed{0.4}
\][/tex]
