Which function represents exponential decay? $$ f(x)=\frac{1}{2}(2)^{x} $$ $$ f(x)=\frac{3}{4}\left(-\frac{1}{5}\right)^{x} $$ $$ f(x)=3\left(\frac{7}{2}\right)^{x} $$ $$ f(x)=2\left(\frac{2}{3}\right)^{x} $$

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An **exponential decay** function has the form:

$$ f(x) = a \cdot b^x $$

where:
- $$ a $$ is a constant (the initial value),
- $$ 0 < b < 1 $$ (the base is between 0 and 1).

This ensures that as $$ x $$ increases, $$ f(x) $$ decreases.

Let's evaluate each of the given functions:

1. **$$ f(x) = \frac{1}{2}(2)^x $$**
   - Base $$ b = 2 $$ (which is greater than 1).
   - This represents **exponential growth**, not decay.

2. **$$ f(x) = \frac{3}{4}\left(-\frac{1}{5}\right)^x $$**
   - Base $$ b = -\frac{1}{5} $$.
   - While $$ |b| = \frac{1}{5} < 1 $$, the negative base introduces oscillations, making it not a standard form for exponential decay.

3. **$$ f(x) = 3\left(\frac{7}{2}\right)^x $$**
   - Base $$ b = \frac{7}{2} $$ (which is greater than 1).
   - This represents **exponential growth**, not decay.

4. **$$ f(x) = 2\left(\frac{2}{3}\right)^x $$**
   - Base $$ b = \frac{2}{3} $$ (which is between 0 and 1).
   - This fits the definition of **exponential decay**.

**Conclusion:**

**$$ f(x) = 2\left(\frac{2}{3}\right)^x $$** represents exponential decay.

**Answer:**  
$$ f(x) = 2\left(\dfrac{2}{3}\right)^{\,x} $$
$$ f(x) = 2\left(\frac{2}{3}\right)^x $$ represents exponential decay.

Which function represents exponential decay? \( f(x)=\frac{1}{2}(2)^{x} \) \( f(x)=\frac{3}{4}\left(-\frac{1}{5}\right)^{x} \) \( f(x)=3\left(\frac{7}{2}\right)^{x} \) \( f(x)=2\left(\frac{2}{3}\right)^{x} \)

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