Use the table to answer the question.

Question

Term Number 	1	2	3	4	5	6
Sequence for
g(x)=20x
20	40	60	80	100	120
Sequence for
f(x)=2x
2	4	8	16	32	64
Zoya is comparing the growth rates of g(x)=20x
 and f(x)=2x
 using their sequences as shown in the table. They notice that the terms of f(x)=2x
 are always less than the terms of g(x)=20x
 and so conclude that f(x)=2x
 grows slower than g(x)=20x
. Where is their mistake in comparing the growth rates?

(1 point)
Responses

Zoya has it backward. The growth rate of f(x)=2x
 is always greater than the growth rate of g(x)=20x
.
Zoya has it backward. The growth rate of  f left parenthesis x right parenthesis equals 2 superscript x baseline  is always greater than the growth rate of  g left parenthesis x right parenthesis equals 20 x .

Zoya must have calculated the growth rates incorrectly. Both functions are growing at the same rate.
Zoya must have calculated the growth rates incorrectly. Both functions are growing at the same rate.

Zoya should be comparing the differences between the values of the terms in each sequence as opposed to just comparing the values of each sequence.
Zoya should be comparing the differences between the values of the terms in each sequence as opposed to just comparing the values of each sequence.

Zoya should have made a bigger table. If they did, they would have seen that the terms of f(x)=2x
 are eventually greater than the terms of g(x)=20x
.

GPT-4o mini · Answer

The correct response is: **Zoya has it backward. The growth rate of f(x)=2^x is always greater than the growth rate of g(x)=20x.** Here's the reasoning: - The function $ f(x) = 2^x $ is an exponential function, which grows much faster than the linear function $ g(x) = 20x $ as $ x $ increases. - While at small values of $ x $ (like 1, 2, 3, etc.), the terms for $ g(x) $ may be greater than those for $ f(x) $, as $ x $ continues to increase, the values of $ f(x) $ will surpass those of $ g(x) $ significantly. - This shows that the exponential growth rate (for $ f(x) $) ultimately outstrips the linear growth rate (for $ g(x) $).