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

Question

. Which statement correctly describes how Austin should do this and what he will observe?(1 point)
Responses

Austin should find where one curve is above the other curve on the graph. The growth rate of g(x)=1.3x
is only greater than the growth rate of f(x)=1.3^x
to the right of their right-most intersection point.
Austin should find where one curve is above the other curve on the graph. The growth rate of g of x is equal to 1 point 3 x is only greater than the growth rate of f of x is equal to 1 point 3 to the x th power to the right of their right-most intersection point.

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^x
Austin should compare the steepness of the curves. The growth rate of g of x is equal to 1 point 3 x will quickly surpass the growth rate of f of x is equal to 1 point 3 to the x th power

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.
Austin should find where one curve is above the other curve on the graph. The growth rate of f of x is equal to 1 point 3 to the x th power is only greater than the growth rate of g of x is equal to 1 point 3 x to the right of their right-most intersection point.

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
Austin should compare the steepness of the curves. The growth rate of f of x is equal to 1 point 3 to the x th power will quickly surpass the growth rate of g of x is equal to 1 point 3 x

Answer 1

The correct statement describing how Austin should compare the growth rates of \( g(x) = 1.3x \) and \( f(x) = 1.3^x \) is:

"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."

This option correctly acknowledges that while \( f(x) = 1.3^x \) (an exponential function) grows more rapidly than \( g(x) = 1.3x \) (a linear function) as \( x \) increases beyond a certain point, the two functions intersect at some points before \( f(x) \) outgrows \( g(x) \) for larger values of \( x \). Thus, examining where one function is above the other, particularly after their last intersection, would provide insight into their growth rates.