To accurately compare the growth rates of the two functions given:
- g(x) = 1.3x: This function represents a linear function, where the growth rate is constant.
- f(x) = 1.3^x: This function represents an exponential function, where the growth rate increases as x increases.
When comparing a linear function (like g(x)) with an exponential function (like f(x)), the exponential function will eventually grow faster than the linear function, as exponential functions grow at an increasing rate while linear functions grow at a constant rate.
Evaluating the statements provided:
A) Incorrect. Although it is true that one curve is above the other at certain points, it does not correctly describe the nature of the growth rates of the given functions.
B) Incorrect. This statement incorrectly identifies that \( f(x) = 1.3^x \) will surpass \( g(x) = 1.3x \) without emphasizing that the exponential function grows faster for larger values of x. However, it implies that \( f(x) \) will catch up quickly, which isn't necessarily accurate for small values.
C) Correct. It appropriately states that there is a point where the growth rate of \( f(x) = 1.3^x \) surpasses \( g(x) = 1.3x \) and acknowledges that this happens after their intersection.
D) Incorrect. This statement incorrectly claims that \( g(x) = 1.3x \) will eventually grow faster, which contradicts the characteristics of linear versus exponential growth.
The correct choice for Austin is:
C) 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.