Given the information that one function (Function #1) is linear and the other (Function #2) is exponential, we can analyze their growth rates:
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Linear functions grow at a constant rate. This means that the increase in the function's value is the same over equal intervals of \( x \).
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Exponential functions grow at an increasing rate. This means that the rate at which the function's value increases itself increases over time, causing the function to rise more steeply than a linear function as \( x \) becomes larger.
From this understanding, we can conclude:
- Initially, the linear function might have a higher rate of growth (especially if it starts at a higher value), but eventually, the exponential function will grow faster and surpass the linear function.
- Therefore, after a certain point (which could be around \( x = 32 \) based on the options provided), the growth rate of the exponential function will be greater than that of the linear function.
Based on that analysis, the correct answer 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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