Since I can't view the image, I can't provide specific details on the graphs of the functions. However, I can explain how to compare the growth rates of linear and exponential functions in general.
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Linear Function (f(x)): A linear function has a constant rate of change. This means that as \( x \) increases, \( f(x) = mx + b \) (where \( m \) is the slope) increases by a fixed amount for each unit increase in \( x \).
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Exponential Function (g(x)): An exponential function grows at a rate that increases over time. Its general form is \( g(x) = a \cdot b^x \) (where \( a \) is a constant and \( b > 1 \)). This means that as \( x \) increases, the value of \( g(x) \) increases much faster than that of a linear function.
Comparison of Growth Rates:
- Initially, a linear function may grow faster than an exponential function for small values of \( x \).
- However, as \( x \) continues to increase, the exponential function will eventually surpass the linear function due to its increasing rate of growth. This phenomenon is often referred to as "exponential growth outpacing linear growth."
Conclusion: In summary, while linear functions have a constant growth rate, exponential functions experience a rapid increase that eventually leads to much higher values as \( x \) becomes large. Thus, if the linear function represents \( f(x) \) and the exponential function represents \( g(x) \), the growth rate of \( g(x) \) will eventually exceed that of \( f(x) \) as \( x \) increases.
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