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The given function is $ f(x) = 100 \cdot 3^x $, which is an exponential growth function. This type of function models situations where the initial quantity (in this case, 100) increases by a constant factor (here, it triples because of $ 3^x $) for each time step.

Let's analyze the options:

A. "The balance of a bank account that began with a $100 deposit with a 3% annual interest" - This would likely be modeled by a function involving compound interest, but the factor would be $ 1.03 $ instead of $ 3^x $, so this doesn't match.

B. "The number of students in a service club that started the year with 100 students and added 3 students each week" - This suggests a linear increase rather than an exponential one, so it doesn't fit the form of the given exponential function.

C. "The number of months that have passed since a fruit fly population that started with 100 fruit flies and tripled each month began growing" - This describes exponential growth, where the population triples each month, matching the function $ 100 \cdot 3^x $.

D. "The number of fruit flies in a population that started with 100 and tripled each month" - This is very similar to C and also matches the exponential growth described by $ 100 \cdot 3^x $.

Thus, the correct answers are **C** and **D**, as both describe exponential growth where the population triples over time, which aligns with the given function.

Would you like further clarification or details on exponential growth? Here are some related questions to explore:

1. How is exponential growth different from linear growth?
2. What is the role of the base in an exponential function like $ 3^x $?
3. How would you solve for $ x $ if given a specific value for $ f(x) $?
4. How would the graph of the function $ 100 \cdot 3^x $ look compared to a linear function?
5. How does compounding in financial applications differ from pure exponential growth?

**Tip:** Exponential growth functions can be identified when quantities multiply by a constant factor per unit of time, which is key in many real-world phenomena like population growth and interest calculations.

Which of the following scenarios could be modeled by the function f^{-1}(x) if f(x) = 100 * 3^x?

Explore a problem involving exponential growth, modeled by the function f(x) = 100 * 3^x. This problem includes analyzing inverse functions and matching real-life scenarios such as population growth. Suitable for students in Grades 9-12.

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