Use the image to answer the question.

Question

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it, each representing a function.

Olive is comparing the growth rates of p(x)=5x
and q(x)=20x
using this graph. Based on the graph, Olive concludes that the growth rate of q(x)=20x
is always greater than the growth rate of p(x)=5x
. Where is her mistake?

(1 point)
Responses

Olive has it backward. The growth rate of p(x)=5x
is always greater than the growth rate of q(x)=20x
.
Olive has it backward. The growth rate of p left parenthesis x right parenthesis equals 5 superscript x baseline is always greater than the growth rate of q left parenthesis x right parenthesis equals 20 x .

Olive is only considering a small portion of the domains of the functions. If she went farther to the left on the graph, she would observe that the growth rate of p(x)=5x
eventually surpasses the growth rate of q(x)=20x
for large negative values of x
.
Olive is only considering a small portion of the domains of the functions. If she went farther to the left on the graph, she would observe that the growth rate of p left parenthesis x right parenthesis equals 5 superscript x baseline eventually surpasses the growth rate of q left parenthesis x right parenthesis equals 20 x for large negative values of x .

Olive is only considering a small portion of the domains of the functions. If she went farther to the right on the graph, she would observe that the growth rate of p(x)=5x
eventually surpasses the growth rate of q(x)=20x
for large values of x
.
Olive is only considering a small portion of the domains of the functions. If she went farther to the right on the graph, she would observe that the growth rate of p left parenthesis x right parenthesis equals 5 superscript x baseline eventually surpasses the growth rate of q left parenthesis x right parenthesis equals 20 x for large values of x .

The growth rate of q(x)=20x
is only greater than the growth rate of p(x)=5x
when x
is greater than approximately 0.

Answer 1

Olive's mistake lies in her misunderstanding of how the growth rates of exponential functions compare, particularly as \( x \) increases.

The correct response would be:

Olive is only considering a small portion of the domains of the functions. If she went farther to the right on the graph, she would observe that the growth rate of \( p(x) = 5^x \) eventually surpasses the growth rate of \( q(x) = 20^x \) for large values of \( x \).

This reflects that \( p(x) = 5^x \) grows exponentially faster than \( q(x) = 20^x \) as \( x \) becomes large, despite \( q(x) \) being bigger at certain intervals or lower values of \( x \).