To identify which function represents exponential decay, we need to understand the general form of an exponential function and the conditions for exponential decay.
An exponential function is typically given by [tex]\( f(x) = a \cdot b^x \)[/tex], where:
- [tex]\( a \)[/tex] is a constant (the initial value)
- [tex]\( b \)[/tex] is the base of the exponential function
- [tex]\( x \)[/tex] is the exponent
For the function to represent exponential decay, the base [tex]\( b \)[/tex] must satisfy [tex]\( 0 < b < 1 \)[/tex]. This ensures that as [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] decreases, which is the characteristic of decay.
Let's analyze each function in the options:
1. [tex]\( f(x) = \frac{1}{2}(2)^x \)[/tex]
- Here, the base [tex]\( b = 2 \)[/tex].
- Since [tex]\( 2 > 1 \)[/tex], this function represents exponential growth, not decay.
2. [tex]\( f(x) = \frac{3}{4}\left(-\frac{1}{5}\right)^x \)[/tex]
- Here, the base [tex]\( b = -\frac{1}{5} \)[/tex].
- The base is negative, and although [tex]\( \left| -\frac{1}{5} \right| < 1 \)[/tex], exponential functions typically require the base to be positive for standard interpretation. This function does not represent a standard form of exponential decay.
3. [tex]\( f(x) = 3\left(\frac{7}{2}\right)^x \)[/tex]
- Here, the base [tex]\( b = \frac{7}{2} \)[/tex].
- Since [tex]\( \frac{7}{2} > 1 \)[/tex], this function represents exponential growth, not decay.
4. [tex]\( f(x) = 2\left(\frac{2}{3}\right)^x \)[/tex]
- Here, the base [tex]\( b = \frac{2}{3} \)[/tex].
- Since [tex]\( 0 < \frac{2}{3} < 1 \)[/tex], this function represents exponential decay.
Therefore, the function that represents exponential decay is:
[tex]\[ f(x) = 2\left(\frac{2}{3}\right)^x \][/tex]
