To determine which statement is true regarding the functions \( f(x) = 100 \cdot 1.5^x \) and \( g(x) = 100x + 100 \), we need to analyze their growth rates.
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Statement #1: While the growth rate of \( f(x) \) is initially greater than the growth rate of \( g(x) \), the growth rate of \( g(x) \) keeps increasing and eventually surpasses the growth rate of \( f(x) \).
- This statement is incorrect because \( f(x) \) is an exponential function which eventually grows faster than any linear function \( g(x) \) as \( x \) increases.
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Statement #2: The growth rate of \( g(x) \) is greater than the growth rate of \( f(x) \) between approximately \( x=0 \) and \( x=4 \).
- This statement is also incorrect. Generally, \( g(x) \) has a constant growth rate of 100 (the slope of the line), whereas \( f(x) \) grows at an increasing rate since it is an exponential function starting from \( f(0) = 100 \) and growing more rapidly as \( x \) increases.
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Statement #3: While the growth rate of \( g(x) \) is initially greater than the growth rate of \( f(x) \), the growth rate of \( f(x) \) keeps increasing and, by \( x=2.5 \), surpasses the growth rate of \( g(x) \).
- This statement is true. At small values of \( x \), \( g(x) \) might appear to be increasing at a faster rate due to its linear nature, but because \( f(x) \) is exponential, it will eventually outpace \( g(x) \) as \( x \) increases, and this point occurs somewhere after \( x=2.5 \).
Therefore, the correct choice is:
Statement #3 is true.