Question

Marshall is comparing the growth rates of f(x)=6(1.5)x
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 approximatel -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 approximatel -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)x
eventually surpasses the growth rate of g(x)=5x+10
for negative 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 negative 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(x)=6(1.5)x
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

Answer 1

Marshall's mistake lies in not considering that exponential functions like \( f(x) = 6(1.5)^x \) grow at a faster rate than linear functions like \( g(x) = 5x + 10 \) as \( x \) increases.

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 reflects the characteristic of exponential growth outpacing linear growth in the long run.