Select the best description of the following exponential equation. \( f(x)=19(3.52)^{x} \) Show your work here Decaying by \( 252 \% \) with every unit of \( x \). Decaying by \( 352 \% \) with every unit of \( x \). Growing by \( 352 \% \) with every unit of \( x \). Growing by \( 252 \% \) with every unit of \( x \).
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To analyze the function \( f(x) = 19(3.52)^{x} \), we can focus on the base \( 3.52 \). Since this base is greater than 1, the function represents exponential growth rather than decay. Next, we can calculate the growth factor by subtracting 1 from the base and converting it to a percentage: 1. Growth factor = \( 3.52 - 1 = 2.52 \) 2. Percentage growth = \( 2.52 \times 100\% = 252\% \) Thus, the function is growing by \( 252\% \) with every unit of \( x \). In conclusion, the correct description is: *Growing by \( 252 \% \) with every unit of \( x \).* Historical Background: Exponential growth concepts can be traced back to the work of mathematicians like Jacob Bernoulli in the 17th century, who studied compound interest and laid foundations for understanding exponential functions. This principle has since been crucial in fields ranging from population dynamics to finance, illustrating how quantities can grow rapidly under certain conditions. Real-World Application: Exponential growth models are especially potent in finance, such as calculating compound interest, where initial investments grow over time. Additionally, they describe phenomena like viral infections or the spread of information on social media, where small changes in the growth rate can lead to drastically different outcomes over time—making understanding these concepts essential for decision-makers.
