To determine the type of function represented by [tex]\( f(x) = 7 \cdot \left( \frac{1}{2} \right)^x \)[/tex], let's analyze its form and properties step by step.
1. Understanding the Function Form:
The given function is [tex]\( f(x) = 7 \cdot \left( \frac{1}{2} \right)^x \)[/tex].
2. Identifying the Components:
In the standard form [tex]\( f(x) = a \cdot b^x \)[/tex], [tex]\( a \)[/tex] is the initial value (constant multiplier), and [tex]\( b \)[/tex] is the base of the exponent.
- Here, [tex]\( a = 7 \)[/tex]
- And [tex]\( b = \frac{1}{2} \)[/tex]
3. Analyzing the Base [tex]\( b \)[/tex]:
- If [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
In this case, [tex]\( b = \frac{1}{2} \)[/tex], which is between 0 and 1. Therefore, the function is of exponential decay type.
4. Confirming the Characteristics:
- Exponential decay functions decrease as [tex]\( x \)[/tex] increases.
- The factor [tex]\( \left( \frac{1}{2} \right)^x \)[/tex] rapidly decreases towards zero as [tex]\( x \)[/tex] increases.
5. Conclusion:
The function [tex]\( f(x) = 7 \cdot \left( \frac{1}{2} \right)^x \)[/tex] clearly fits the description of an exponential decay function.
Thus, the correct answer is B. Exponential decay.
