Select the term that best represents each formula. a. \( h(t)=\frac{3}{2}(2)^{t} \) b. \( f(x)=0.75^{x} \) Justify your response for (a) and (b). (b)

To categorize each of the given formulas, we'll identify whether they represent exponential growth or exponential decay based on their structure. --- **a. \( h(t) = \frac{3}{2} \cdot 2^{t} \)** - **Term:** **Exponential Growth** **Justification:** - The general form of an exponential function is \( f(t) = a \cdot b^{t} \), where: - \( a \) is the initial value. - \( b \) is the base that determines the growth or decay. - In \( h(t) = \frac{3}{2} \cdot 2^{t} \): - The base \( b = 2 \) is greater than 1. - A base greater than 1 signifies that the function is **exponentially increasing** as \( t \) increases. - Therefore, \( h(t) \) represents **exponential growth**. --- **b. \( f(x) = 0.75^{x} \)** - **Term:** **Exponential Decay** **Justification:** - Using the same general form \( f(x) = a \cdot b^{x} \): - Here, \( a = 1 \) (implicitly) and \( b = 0.75 \). - The base \( b = 0.75 \) is between 0 and 1. - A base between 0 and 1 indicates that the function decreases as \( x \) increases. - Therefore, \( f(x) \) represents **exponential decay**. --- **Summary:** - **a.** Exponential Growth - **b.** Exponential Decay