To analyze the growth rates of the two functions given that \( f(x) \) is linear and \( g(x) \) is exponential, we can consider the characteristics of each type of function.
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Linear Function \( f(x) \): A linear function can be represented as \( f(x) = mx + b \), where \( m \) is the slope. The growth rate of a linear function is constant, meaning it increases by the same amount regardless of the value of \( x \).
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Exponential Function \( g(x) \): An exponential function can be represented as \( g(x) = a \cdot b^x \), where \( a \) and \( b \) are constants and \( b > 1 \). The growth rate of an exponential function increases rapidly as \( x \) increases. In fact, the rate of growth accelerates the larger \( x \) becomes.
Comparison of Growth Rates
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Initially (for small values of \( x \)), a linear function can have a greater growth rate depending on its slope. However, as \( x \) increases, the exponential function's growth rate quickly outpaces the constant rate of the linear function.
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At some point (like \( x = 32 \) in your scenario), the exponential growth will surpass the linear growth rate.
Conclusion
Given this information, the correct response would be:
"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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