Which function represents exponential decay? \( f(x)=\frac{1}{2}\left(\frac{3}{2}\right)^{x} \) \( f(x)=\frac{1}{2}\left(-\frac{3}{2}\right)^{x} \) \( f(x)=4\left(-\frac{2}{3}\right)^{x} \) \( f(x)=4\left(\frac{2}{3}\right)^{x} \)

To identify which function represents exponential decay, we need to analyze each option based on the general form of an exponential function: \[ f(x) = ab^x \] - **Exponential Growth:** Occurs when the base \( b > 1 \). - **Exponential Decay:** Occurs when the base \( 0 < b < 1 \). - **Negative Bases:** Typically lead to alternating signs, which is not characteristic of standard exponential growth or decay functions. Hence, we'll focus on functions with positive bases. Let's examine each function: 1. **\( f(x) = \frac{1}{2}\left(\frac{3}{2}\right)^x \)** - **Base:** \( \frac{3}{2} \) (which is greater than 1) - **Behavior:** Exponential growth 2. **\( f(x) = \frac{1}{2}\left(-\frac{3}{2}\right)^x \)** - **Base:** \( -\frac{3}{2} \) (negative) - **Behavior:** Alternates in sign; not a standard exponential decay 3. **\( f(x) = 4\left(-\frac{2}{3}\right)^x \)** - **Base:** \( -\frac{2}{3} \) (negative) - **Behavior:** Alternates in sign; not a standard exponential decay 4. **\( f(x) = 4\left(\frac{2}{3}\right)^x \)** - **Base:** \( \frac{2}{3} \) (between 0 and 1) - **Behavior:** Exponential decay **Conclusion:** The function that represents exponential decay is: \[ f(x) = 4\left(\frac{2}{3}\right)^x \] **Answer:** \( f(x)=4\left(\frac{2}{3}\right)^{x} \)