To compare the growth rates of the two functions, we note that Function #1 is linear (represented by the dotted straight line) and Function #2 is exponential (represented by the increasing solid curve).
- Linear functions grow at a constant rate. The slope (or growth rate) remains the same regardless of the value of x.
- Exponential functions grow at an increasing rate. As x increases, the growth rate of an exponential function accelerates.
Given this understanding:
- Initially, linear functions can grow faster than exponential functions if they start at a higher value or at the same value but have a steeper initial slope.
- However, as x increases, exponential functions will eventually surpass linear functions in terms of growth rate.
Thus, for this problem, it is implied that there would be a point (in this case, at x = 32) where the exponential growth of Function #2 would eventually outpace the linear growth of Function #1.
The most accurate response based on this analysis is:
Initially, Function #1 has the greater growth rate. After x = 32, the growth rate of Function #2 surpasses the growth rate of Function #1.
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