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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 .

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 .

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 .

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
.

Answer 1

Marshall's mistake lies in his assessment of the growth rates of the two functions.

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⋅1.5x eventually surpasses the growth rate of g(x)=5x+10 for large values of x.

This conclusion is supported by the fact that while \( g(x) \) (a linear function) grows at a constant rate, \( f(x) \) (an exponential function) increases at an accelerating rate as \( x \) becomes larger. Thus, for sufficiently large values of \( x \), \( f(x) \) will outperform \( g(x) \).