Austin is using graphs to compare the growth rates of g (x) = 1.3x and f(x) = 1.3*

Question

. Which
statement correctly describes how Austin should do this and what he will observe? (1 point)
Austin should compare the steepness of the curves. The growth rate of f(x) = 1.3* will
quickly surpass the growth rate of g (x) = 1.3x.
Austin should compare the steepness of the curves. The growth rate of g (x) = 1.3x will
quickly surpass the growth rate of f(x) = 1.3*
Austin should find where one curve is above the other curve on the graph. The growth rate of
9 (x) = 1.3x is greater than the growth rate of f(x) = 1.3* between the intersection points of
the curve.
Austin should find where one curve is above the other curve on the graph. The growth rate of
f (x) = 1.3* is only greater than the growth rate of g (x) = 1.3x to the right of their right-most
intersection point.

Answer 1

The correct statement is:

Austin should compare the steepness of the curves. The growth rate of f(x) = 1.3^x will quickly surpass the growth rate of g(x) = 1.3x.

This is because \( f(x) = 1.3^x \) is an exponential function, while \( g(x) = 1.3x \) is a linear function. Exponential functions grow at a faster rate than linear functions as \( x \) increases. Therefore, Austin will observe that the growth rate of the exponential function \( f(x) \) surpasses that of the linear function \( g(x) \) as \( x \) becomes sufficiently large.