To correctly analyze and compare the growth rates of the functions \( g(x) = 1.3^x \) (exponential growth) and \( f(x) = 1.3x \) (linear growth), we need to consider how these two functions behave as \( x \) increases.
The function \( g(x) = 1.3^x \) grows exponentially, while \( f(x) = 1.3x \) grows linearly. As \( x \) becomes very large, exponential functions will eventually surpass linear functions in growth rate due to their nature.
Now, let's analyze the provided options:
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Option a suggests that the growth rate of \( f(x) = 1.3^x \) will surpass \( g(x) = 1.3x \), which is incorrect.
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Option b suggests the opposite, that \( g(x) = 1.3x \) will surpass \( f(x) = 1.3^x \), which is also incorrect since \( 1.3^x \) will ultimately surpass \( 1.3x \).
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Option c accurately states that \( f(x) = 1.3^x \) is greater than \( g(x) = 1.3x \) to the right of their right-most intersection point. This is true since \( 1.3^x \) will eventually grow faster than \( 1.3x \).
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Option d incorrectly states that the growth rate of \( g(x) = 1.3x \) is greater than \( f(x) = 1.3^x \) between the intersection points, which is misleading since, after a certain point, the exponential function dominates.
Therefore, Option c is the correct response:
Austin should find where one curve is above the other curve on the graph. The growth rate of \( f(x) = 1.3^x \) is only greater than the growth rate of \( g(x) = 1.3x \) to the right of their right-most intersection point.
2 answers