Let's examine each function to determine which one represents exponential decay.
1. \( f(x) = \frac{1}{2} (2)^x \)
- Here, the base of the exponent is 2, which is greater than 1. In an exponential function \( a(b)^x \), if \( b > 1 \), the function represents exponential growth.
- Therefore, this function represents exponential growth, not decay.
2. \( f(x) = \frac{3}{4} \left(-\frac{1}{5}\right)^x \)
- This function involves a negative base \(-\frac{1}{5}\). Exponential functions typically have a positive base to ensure consistent growth or decay properties across the domain of real numbers.
- Since having a negative base complicates the behavior of the function, especially for non-integer exponents, this does not typically represent exponential decay in the conventional sense.
3. \( f(x) = 3 \left(\frac{7}{2}\right)^x \)
- Here, the base of the exponent is \(\frac{7}{2}\), which is greater than 1. This means the function represents exponential growth.
- Therefore, this function represents exponential growth, not decay.
4. \( f(x) = 2 \left(\frac{2}{3}\right)^x \)
- Here, the base of the exponent is \(\frac{2}{3}\), which is a fraction between 0 and 1. In an exponential function \( a(b)^x \), if \( 0 < b < 1 \), the function represents exponential decay.
- Thus, this function represents exponential decay.
After carefully examining each option, the function that represents exponential decay is:
[tex]\[ f(x) = 2 \left(\frac{2}{3}\right)^x \][/tex]
So, the correct answer is:
[tex]\[ f(x) = 2 \left(\frac{2}{3}\right)^x \][/tex]
