An exponential decay function generally has the form \( f(x) = A \cdot b^x \), where \( 0 < b < 1 \). Here, \( b \) is the base of the exponential function which dictates the decay, and \( A \) is a constant that can stretch or compress the function vertically. A stretch occurs when the constant \( A \) is greater than 1.
Let's analyze each given option to check for the conditions of an exponential decay function and determine which one represents a stretch.
1. \( f(x) = \frac{1}{5}\left(\frac{1}{5}\right)^x \)
- The base is \( \frac{1}{5} \), which is between 0 and 1. Therefore, it represents exponential decay.
- The constant \( A \) is \( \frac{1}{5} \), which is less than 1.
- This is not a stretch.
2. \( f(x) = \frac{1}{5}(5)^x \)
- The base is \( 5 \), which is greater than 1. This does not represent exponential decay.
- Hence, we discard this option as it is not even an exponential decay function.
3. \( f(x) = 5\left(\frac{1}{5}\right)^x \)
- The base is \( \frac{1}{5} \), which is between 0 and 1, indicating exponential decay.
- The constant \( A \) is \( 5 \), which is greater than 1.
- This represents a stretch of an exponential decay function.
4. \( f(x) = 5(5)^x \)
- The base is \( 5 \), which is greater than 1. This does not represent exponential decay.
- Thus, we discard this option as it is not an exponential decay function.
Based on the analysis, the correct option is:
[tex]\[ f(x) = 5\left(\frac{1}{5}\right)^x \][/tex]
This function meets both criteria for being a stretch of an exponential decay function. The base \( \frac{1}{5} \) is between 0 and 1, indicating decay, and the constant \( 5 \) is greater than 1, indicating a vertical stretch. Therefore, the answer is:
Option 3
