Question

Use the image to answer the question.

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it, corresponding to two functions.

Marshall is comparing the growth rates of f(x)=6⋅1.5x
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⋅1.5x
. Where is his mistake?

(1 point)
Responses

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

The growth rate of g(x)=5x+10
is only greater than the growth rate of f(x)=6⋅1.5x
when x
is greater than approximately −1.3
.
The growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 is only greater than the growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline when x is greater than approximately negative 1.3 .

Marshall has it backward. The growth rate of f(x)=6⋅1.5x
is always greater than the growth rate of g(x)=5x+10
.
Marshall has it backward. The growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline is always greater than the growth rate of g left parenthesis x right parenthesis equals 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⋅1.5x
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 left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline eventually surpasses the growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 for large values of x .

Answer 1

Marshall's mistake is that he is only considering a limited portion of the domains of the functions and not recognizing the long-term behavior of the exponential function compared to the linear function.

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 response accurately captures the fact that exponential functions will eventually overtake linear functions as x increases, despite the linear function having a higher value at smaller x values.