Marshall is comparing the growth rates of f(x)=6(1.5)^x

Question

and g(x)=5x+10
using this graph. Based on the graph, he concludes that the growth rate of g(x)=5x+10
is always greater than the growth rate of f(x)=6(1.5)^x
. Where is his mistake?

(1 point)
Responses

The growth rate of g(x)=5x+10
is only greater than the growth rate of f(x)=6(1.5)^x
when x is greater than approximately -1.3
The growth rate of g of x is equal to 5 x plus 10 is only greater than the growth rate of f of x is equal to 6 times 1 point 5 to the x th power when x is greater than approximately -1.3

Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6(1.5)6x
eventually surpasses the growth rate of g(x)=5x+10
for large values of x.
Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f of x is equal to 6 times 1 point 5 to the x th power eventually surpasses the growth rate of g of x is equal to 5 x plus 10 for large values of x.

Marshall is backward. The growth rate of f(x)=6(1.5)^x
is always greater than the growth rate of g(x)=5x+10
Marshall is backward. The growth rate of f of x is equal to 6 times 1 point 5 to the x th power is always greater than the growth rate of g of x is equal to 5 x plus 10

Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6(1.5)^x
eventually surpasses the growth rate of g(x)=5x+10
for negative values of x.

Answer 1

Marshall's mistake lies in his understanding of the growth rates of the two functions over their entire domains. The correct response is:

Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6(1.5)^x eventually surpasses the growth rate of g(x)=5x+10 for large values of x.

This means that while \(g(x)\) may initially be greater for certain values of \(x\) (especially for smaller \(x\)), the exponential function \(f(x)\) will eventually grow at a faster rate and surpass \(g(x)\) as \(x\) increases. Thus, he is not recognizing the long-term behavior of the exponential function compared to the linear function.